Logarithm Fundamentals
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Back to the basics with logarithms. Full playlist: https://www.youtube.com/playlist?list...Home page: https://www.3blue1brown.comBrought to you by you: https://3b1b.co/ldm-thanksTriangle of power
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welcome back to lockdown math today we
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are going to be talking about logarithms
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and kind of a back to the basic sort of
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lesson the intent is for someone who
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maybe is not familiar with logarithms
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yet or who has seen some of the rules
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but has been confused by them and as
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always to kick things off I just want to
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get a sense of where the audience is at
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right now because I have a couple
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suspicions but I think doing a live poll
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to see where everyone is might be
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helpful so if you can go to 3 B 1 B Co
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/live which you can find the link in the
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description and you can also just follow
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the link that's up on screen here you'll
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find a poll and at the moment it asks
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what best describes your relationship
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with logarithms a I've never heard about
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them before we're never learned about
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them before B I've learned about them
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but sometimes get confused by all of the
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properties see I understand them but
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wouldn't know how to teach them and D I
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understand them well and could
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comfortably teach them to someone else
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to make them understand well too so
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we've got a good split this is good and
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interesting like I said the intent for
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this is to create a lesson that I can
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point people to in the future if they're
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just not comfortable with logarithms and
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I want to be able to say oh here's a
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place that you can go for how I think
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you know how I think you could approach
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it intuitively and for those who are in
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the camps of let's say understanding it
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but not knowing how to teach it I think
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there's an interesting meta question
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around logarithms because I was
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scrolling around a couple teacher forums
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before doing this particular lecture and
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when people ask what is the hardest
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topic to teach in high school math in
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the sense that students seem to have
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trouble with it the most logarithms is
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one of the most commonly indicated
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answers which is interesting and I can
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guess maybe it's because there's a ton
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of these properties that you end up
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having to learn you know so if we skip
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ahead of that where we're gonna go
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you've got all these piles of rules that
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just look like a bunch of algebra that
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can be hard to remember and easy to kind
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of mix things up in your head and I
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think when people have you know
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sort of nightmarish recollections of
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what high school math was like and what
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logarithms did for them it's often those
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particular formulas coming to mind and
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what I want to do today is try to talk
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through one how to think about them but
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also just on the meta level of if you're
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teaching somebody algebra what are the
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points worth emphasizing what's the way
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to get it built in their intuitions so
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let's go ahead and see how people are
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answering on this one we've got a pretty
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even split between three three possible
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categories all right so the most common
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viewer right now answered see that they
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understand them but they wouldn't know
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how to teach them the second most common
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are those who said they understand them
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well and they could comfortably teach
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them so this is sort of revealing that
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okay the existing audience on the
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channel at the moment that's people who
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are into math and they're going to be
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pretty comfortable with logarithms and
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then after that be people who are
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confused by some of the properties and
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then at the bottom people have never
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learned about them now if you watching
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are one of the people who answered a
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that you've never learned about them or
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that you're not comfortable keep in mind
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this lecture is actually meant for you
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so when we're doing some live quizzing
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and we're gonna have an answer I'm here
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and we're gonna have people you know
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answering probably pretty quickly
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probably correctly given that as you can
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see most of them are coming in
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comfortable with logarithms don't let
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that be intimidating don't let that be
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something that indicates you should be
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answering quickly or you should
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necessarily be getting the right answer
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right so I think the most intuitive way
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to think about logs if you're just going
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to start off is to say at a very high
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level logarithms are something that can
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take in a power of 10 and it just spits
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out the number of zeros at the end of
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the number it does more than that as
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we'll talk about but just at the very
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high level this is getting a sense for
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what they do
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and I think it's pretty important
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because what you're saying here is hey
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look as we walk from one to ten to a
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hundred to a thousand to ten thousand
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each step were multiplying by ten the
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output of the log the log of that value
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just steps up in these increments of one
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so you can think about this as saying
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hmm when I plug in a number to the log
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if that number happens to be a power of
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10 I'm just counting the number of zeros
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what's log of a thousand oh it has three
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zeros on it what's log of a million I
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would
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six zeros on it it's the zero counting
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function and I think you can get
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yourself very far in terms of
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understanding very various properties of
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logarithms just by thinking in the back
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of your mind okay when I plug in a power
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of 10 it's a zero counting function i
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plug in a billion how many zeros are at
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the end of a billion and these numbers
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are all very intuitive to us we see them
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in the news we we anchor our intuitions
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in terms of how many digits does a
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number have and logarithms are just
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capturing that idea and in fact
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logarithms have been a particular
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interest in the last couple of months
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what's kind of interesting is if you
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look at this Google Trends chart that's
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just looking for logarithmic scale okay
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people searching for the term
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logarithmic scale you'll notice you know
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there's kind of an ambient interest
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sitting around in September of 2019 up
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to forever of 2020 and for some reason
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suddenly there is a strong interest in
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understanding logarithmic scales from
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the general public I wonder why that is
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and we see this sitting around early
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March or so and of course this is
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because this is when the corona outrace
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outbreak was really starting to kick
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into high gear and everyone wanted to
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understand exponential growth and a
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common way that exponential growth is
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plotted is with what's known as a
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logarithmic scale so I actually made a
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video about this and in it I was you
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know creating some animations and wanted
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to illustrate this idea of exponential
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growth and the main idea here I'll go
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ahead and skip back to a different
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animation is if you're tracking the
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numbers in this case this was the number
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of recorded cases of kovat 19 outside of
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mainland China in the months leading up
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to March you could just track what the
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absolute number is but the pattern that
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you'll find is that as you go from one
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day to the next you tend to be
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increasing multiplicatively
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it's a little bit like earlier we were
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seeing the powers of 10 one step to the
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next you're multiplying by some amount
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the way that the virus was growing was
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very similar from one day to the next
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you're multiplying not quite by a
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constant but in this case for this
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sequence of days it was around 1.2 in
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that region you're multiplying by
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something so when you're plotting this
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it ends up looking like this you know
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classic exponential curve that curves
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upward and I can sometimes make it hard
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to see where it's going or what the
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overall
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pattern is so a common trick is to say
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instead of looking at this y-axis that
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increases linearly as in here I'm going
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from 5k to 10k 10k to 15 K 15 K to 20 K
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each step is additive we're adding five
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thousand instead use a y-axis where each
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step is multiplicative so you're going
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from 10 to 100 100 to 1000 1000 to 10 10
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thousand all of these are increases by
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multiplying by 10 and what you could say
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is the y-axis is now plotting not the
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total number of cases but the logarithm
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of the total number of cases and this
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actually makes it kind of easier to see
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on a plot if you wanted to project out
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what that trend would do and you know
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it's a little bit of a naive model to
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say oh it's going to grow exactly
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exponentially but in the early phases of
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something like this that is what it is
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so I kind of fast forward in the
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animation I made for that video and
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what's interesting is if back then I
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think I posted it on March 6th if you
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just found a line of best fit and you
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stretched it out and you said when is
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that line going to cross a million which
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because the y-axis is growing but with
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multiplicative steps each time that you
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step up you're multiplying by 10 so even
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if it might seem like the 20,000 cases
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or so that it was back then is very far
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from a million you know when you're
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understand logarithmic scales it
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actually didn't seem that far it was
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only 30 days away
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if you naively just drew out that line
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and in fact fast forward to around April
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5th which is when that would have
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predicted we hit a million cases outside
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China that's pretty much the day that it
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happened
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I think plus or minus a day but I don't
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remember exactly but it was right in
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that neighborhood because I remember
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thinking wow it was kind of a naive
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model for the video to even use and it's
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shocking that it matched so exactly
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thankfully since then the growth has
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stopped being exponential so if you look
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at it on a logarithmic plot instead of
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going up in a straight line that starts
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to taper off but point being anytime
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that you're coming across something in
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nature or even in a man-made construct
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where what's natural to think about are
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multiplicative increases logarithms come
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in to help you so let's go ahead and
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think about what these actually are how
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are they defined okay
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and actually there was a question asked
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on Twitter right before we started that
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I think hits this very perfectly so max
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182 asks us additions inverse is
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subtraction multiplication inverse is
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division but I never truly understood if
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exponentiations inverse was n through
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ting or logarithms if either one could
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be you know if either one could really
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be called that way can they that is such
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a fantastic question max and I think it
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comes down to the fact that with
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addition and multiplication you're not
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or I'll just draw it out for you
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actually this is gonna be easiest if we
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have some kind of exponential
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relationship let's say 10 to the power 3
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is equal to 1000 there's three different
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numbers at play here we're showing a
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relationship between the 10 the 3 and
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the 1,000 and aside from writing it with
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an exponent there's two other ways we
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could write that same relationship we
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could also say that the cube root of
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1,000 is equal to 10 this is asking a
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thousand what number raised to the third
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is equal to one thousand that's sort of
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what the cube root is asking and another
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way that you could raise the exact same
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thing is to say the log base 10 of 1000
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is equal to three okay three different
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notations the same exact relationship a
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while ago I made this video about an
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alternate possible notation that would
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Center in the idea that you think of
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this relationship between the three
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numbers with a triangle where you'd have
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our 10 sitting down here the power
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sitting at the top 10 to the third and
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the thing they equal sitting to the
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lower right and whenever you want to
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talk about a function or an operation
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between two of these numbers you
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indicate which one of them you're
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leaving out so to write 10 cubed we
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would include the 10 on the lower left
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the 3 on the upper right and then we
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leave out that bottom one so this is
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indicating that we're taking a power 10
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10 cubed for that radical what you would
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say is we know what's on the bottom
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right something to some power equals a
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thousand we even know what the power is
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so
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than cubed equals that thousand but the
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thing we don't know is on the bottom
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left that is the sense in which radicals
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are an opposite of exponentiation where
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if the thing you don't know instead of
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being on that bottom right is on the
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bottom left
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but what logarithms are it's an inverse
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in another sense because what it's
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saying is in this triangle relationship
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we know the base it's ten we know the
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power that it should be a thousand but
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the thing we don't know is what's in
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that exponent so to answer your question
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maybe even more briefly we could say
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that 10 to the X the inverse of that the
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inverse is the log base 10 you know of Y
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of some other variable whereas if you're
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taking X to the tenth some unknown
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raised to a power the inverse of that is
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going to be the tenth root of some other
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value and on the triangle it's basically
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asking which of the things do we
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consider to be a variable so are you
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considering that lower right to be a
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variable quantity are you considering
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that top to be a variable quantity and
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what what is the unknown but I really
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liked this idea of making explicit how
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we have three totally different
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notations for the same exact fact one of
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them you're using relative positions of
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the numbers one of them we introduced a
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new symbol this radical and one of them
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we introduced a new word log so these
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three syntactically different ways to
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communicate the same idea seemed wrong
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and so I made this video about you know
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an alternate possible notation and while
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I don't necessarily think that oh we
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should teach logarithms with this
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triangle because convention is what it
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is so it's better to start getting
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people used to the usual expression what
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I do like about it and starting off with
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it is when you see and think about this
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triangle it's really emphasizing that
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what the log wants to be is that
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exponent every time that you see log of
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some value you should think in your mind
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okay whatever this number is it really
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wants to be an exponent it wants to be
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an exponent and we'll see more of what
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that means as we go on
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okay so every time you see a log it
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wants to be an exponent this value three
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and more specifically it should be an
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exponent sitting on top of whatever that
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base is now in terms of convention for
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the first part of this video I'm just
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going to be using log without a base
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written on it to be the shorthand for
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log base 10 because log base 10 will be
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the most intuitive thing out there you
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should know that often in math the
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convention instead is that log without
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anything might mean log base E
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there's also another notation for that
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Ln for natural log we're going to talk
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all about the natural log next time so
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don't worry too much about that right
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now
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and there's also yet another convention
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often if you're in a computer science
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setting log without any added sugar to
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indicate what it is defaults to meaning
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log base two so this can sometimes be a
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source of confusion but it basically
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depends on what discipline you're in in
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math not math math people really like a
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base of e we'll see why next lecture in
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I don't all say engineering but really
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it's anything where you want good
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intuition with our normal base ten
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number system log means log base ten and
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if you're curious often in computer
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science settings log base two comes up
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all the time so like I said in the back
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of your mind if you're trying to think
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of some of these properties just resting
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on the idea that log counts the number
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of zeros at the end of a number that can
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get you a really far away so we're going
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to start going through a couple of these
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properties and I want to do this just
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with a set of practiced examples so
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we'll transition away from the pole and
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this time to the first proper question
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and the question asks you which of the
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following is true a the log of a
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thousand times s cubed C log of a
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thousand times x equals three plus the
20:25
log of X D log of a thousand times x
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equals three to the power of log of X
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and E none of the above okay and
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remember like I said earlier we should
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fully expect that all of those people at
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the beginning who said they understand
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logs well they're going to be answering
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immediately they're going to be
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answering correctly
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but if you're someone who doesn't don't
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let that intimidate you when you're
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looking at a problem like this one what
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I would encourage you to do is just plug
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in various powers of ten and think in
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terms of the idea that the log function
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counts the number of zeros so I'll give
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you a little moment to think about that
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[Music]
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[Music]
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great okay so I'll go ahead and grade it
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and as always if that's faster than what
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you're comfortable with know that it's
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only because I want to proceed forward
21:44
with the lesson so in this case the
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correct answer comes out to be log of a
21:50
thousand times X is the same as taking
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three plus the log of X okay and now
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let's think about that for a moment and
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like I said when you're just getting
21:57
started with them I think the best thing
21:59
to do is just be comfortable plugging in
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various numbers and the best numbers to
22:02
plug in are the ones that are already
22:04
powers of 10 so if you're asking
22:07
something like log of a thousand times X
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well I don't know let's just plug in
22:13
something for X log of a thousand times
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a hundred okay well we know how many
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zeros are going to be in the final
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answer here well a thousand times 100 is
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a hundred thousand we already
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intuitively have this idea that when we
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multiply two powers of 10 we're just
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taking the zeros the three zeros from
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that thousand the two zeros from that
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hundred and we're putting them next to
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each other so it should be five total
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zeros but if you really reflect not just
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on how did the number turn out but why
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did it turn out that way it was the
22:42
three zeros from that thousand plus the
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two zeros from what that 100 which we
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could also write by saying the number of
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zeros in a thousand plus the number of
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zeros in 100 so this idea that a
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logarithm of the product of two things
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is the sum of the logarithms of those
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two things in the context of powers of
23:05
ten that's just communicating what's
23:06
already a super intuitive idea for a lot
23:08
of us if you take two powers of ten and
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you multiply them you just take all of
23:11
their zeros and kind of cram them on to
23:13
each other so the way I've written
23:14
things out here it's actually indicative
23:16
of a slightly more general fact which is
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going to be our very first property of
23:20
logarithms which is that if we take the
23:23
log of a times B it equals the log of a
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plus the log of B and now anytime you
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see one of these logarithm rules if you
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find yourself you know squinting your
23:32
eyes or a little bit I'm confused by how
23:34
to remember it just plug in examples I'm
23:37
being redundant I'm saying this a lot
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but it's because I think it's very easy
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to forget once you're swamped in the
23:42
algebra itself and you're sitting on
23:43
some kind of test and it's just got a
23:44
lot of symbols to remind yourself you
23:47
our okay to just plug in some numbers
23:49
that's a fine thing to do and often it's
23:52
a great way to yield intuition so in
23:54
this case saying log of a times B and
23:57
breaking it apart we could just think oh
23:59
that log of a hundred times a thousand
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which is five there's five zeros in it
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breaks up in terms of the number of
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zeros in each given part great wonderful
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so carrying that intuition further let's
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try another practice problem and again
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if you know it great you'll be able to
24:16
answer it fine but maybe think not just
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what is the answer but how would I
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explain this answer to someone or how
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would I try to get a student to come to
24:24
this answer on their own without me
24:26
having to tell them what the answer is
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and so there's two potential audience
24:30
members there are those who are
24:31
interested in the lesson itself and then
24:33
those who are interested in the meta
24:34
lesson so a question asks again which of
24:38
the following is true a log of X to the
24:41
N is equal to n times log of X B log of
24:45
X to the N is equal to log of X to the
24:48
power n C log of X to the N is equal to
24:52
n plus log of X or D log of X to the N
24:56
is equal to log of n times log of X or
24:59
none of the above so again take a moment
25:04
to answer as people do answer we'll
25:05
start to see them roll in and if you are
25:09
struggling to think of where to even
25:11
start just plug in some powers of 10 and
25:13
see what intuition you might get from it
25:24
[Music]
25:40
you
25:58
[Music]
26:05
you
26:10
all right so it looks like just about
26:12
everybody is landing on one particular
26:15
answer hopefully it's the right answer
26:17
like I said all to be expected so let's
26:19
go ahead and see if our majority ended
26:22
up correct and they did so the correct
26:24
answer here is a which it looks like
26:26
4000 of you got congratulations
26:29
telling us that log of X to the power n
26:31
is equal to n times log of X so again
26:35
let's say that you're trying to teach
26:37
this to someone or if you're trying to
26:39
come to grips with what it means
26:40
yourself I think a fine place to start
26:42
is plugging something in and in this
26:46
case for log of X to the power n let's
26:49
just try it with a hundred to the power
26:51
three and you can try it with other ones
26:52
to see if the patterns you're doing
26:54
actually work but if you're thinking it
26:56
through not in terms of simply seeing
26:58
what the answer is but trying to think
26:59
of why the answer turned out that way
27:01
sometimes one example will do because a
27:04
hundred cubed we can think of that as
27:06
taking well that's three copies of 100
27:09
I'm taking three copies of 100 and when
27:13
I multiply all that out and I think of
27:15
log is counting the number of zeros when
27:17
we say oh it's going to be some number
27:18
that just has six zeros on it that's
27:20
what it means to take a hundred times
27:21
100 times 100 I can just think of
27:24
grouping all of those zeros together to
27:26
get a million so this number is going to
27:29
be six but if we think actually why was
27:32
it six not just that's the number of
27:33
zeros inside the million where that six
27:35
came from is that we had three copies of
27:39
that 100 and each of those 100 had two
27:42
different zeros see that way it's a more
27:44
general way you can think about it where
27:46
if instead of taking a hundred cubed we
27:47
were looking at a thousand cubed or a
27:49
thousand to the N or X to the power n
27:52
you can think that it's whatever that
27:54
value of n was the number of copies we
27:56
were multiplying in times the number of
27:59
well let's see it's not x times the
28:02
number of zeros they were in whatever we
28:04
substituted for X which in this case was
28:06
100 so if instead I had taken something
28:09
like log of 10,000 to the power n this
28:14
would be the same as taking n copies of
28:18
that thousand ten thousand counting the
28:20
number of zeros in each one of them we
28:22
for so it would be n times four and of
28:27
course the general property that most of
28:29
you correctly answered is that you have
28:30
this lovely little effect where when you
28:32
see the log of something raised to a
28:33
power that little power hops down in
28:36
front of it and you just have log of
28:39
what was on the inside now one of the
28:41
maybe most important implications of
28:45
that I don't know if you'd call it an
28:47
implication or if you'd call it every
28:49
statement of the definition if I'm
28:51
taking log and I'll just reinforce eyes
28:53
its base 10 of 10 to the power n we can
28:58
kind of think of that little n is
28:59
hopping down in front and becomes n
29:01
times the log base 10 of 10 which is of
29:06
course 1 this expression you can think
29:10
of as either counting the number of
29:11
zeros at the end or more generally it's
29:13
asking 10 to the what equals 10 and the
29:16
answer is simply 1 which is very
29:19
reassuring because another way that you
29:21
could go back and just read this
29:22
original expression is saying 10 to the
29:25
what equals 10 to the N oh well if the
29:27
answer is n okay now with every given
29:31
logarithm property that we have so in
29:33
this case we just found one log of X to
29:35
the power n involves that n hopping in
29:37
front there's always going to be a
29:38
mirror image exponential property and
29:41
that's another way that we can help to
29:42
get ourselves a little bit of intuition
29:44
for these so let me just cover up some
29:47
of the future properties we're going to
29:48
get to here try to hide where we're
29:52
going what we just found raising
29:55
something to the end that hops in front
29:56
this corresponds to the exponential
29:59
property that if I take 10 to the X and
30:01
raise that whole thing to the power n
30:03
that's the same as taking 10 to the N
30:05
times X and this corner just gets us to
30:08
another intuition that you might have
30:10
for logarithms which is they kind of
30:13
they're like exponentiation turned
30:15
inside out and here's what I mean by
30:17
that the things sitting on the inside of
30:20
the log if I'm taking log of a you
30:22
should be thinking of that as the whole
30:23
outer expression for something that's
30:27
exponential in this case the a the thing
30:29
on the inside corresponds to 10 to the X
30:32
the the output of the function
30:34
whereas the entire thing itself the log
30:38
of a corresponds to what's on the inside
30:42
over here just what's the what's the
30:46
exponent of the ten so wherever you see
30:49
a log expression here you should be
30:50
thinking that plays the role of an
30:51
exponent on the right side and every
30:53
time you see a exponential the entire 10
30:57
to the X expression the whole outer
30:59
component on the right side that
31:01
corresponds to something that's sitting
31:03
on the inside of one of the logs okay
31:06
and we saw this above right the idea
31:08
that when we're multiplying on the
31:09
inside that's adding on the outside well
31:12
if logs kind of turn Exponential's
31:14
inside out that's telling us that
31:15
multiplying on the outside multiplying
31:18
the outputs of the function is the same
31:20
as adding on the inside because each of
31:22
these logs like log a and log B is
31:24
playing the role of the X and the y in
31:27
the expression on the right okay so with
31:30
that let's keep playing let's just do a
31:32
couple more of these and see how many of
31:33
these properties that we can build up an
31:35
intuition for so this last one
31:37
very nice thinking of exponents hopping
31:39
down the next one is something that
31:42
might look a little bit weird to those
31:45
who are not necessarily familiar with
31:46
logarithms but again plug in some
31:48
numbers to gain some intuition for it
31:50
and we'll give it a little bit more a
31:52
moment to pull up which of the following
31:55
is true log base a of B is negative log
31:59
base B of a okay log base a of B is one
32:03
divided by log base B of a log base a of
32:06
B is one minus log base B of a log base
32:10
B of A is log base a of 1 divided by B
32:12
or none of the above so it's asking what
32:16
happens when we swap the base with
32:18
what's sitting inside of the logarithm
32:20
and I'll just give you a minute or two
32:23
to answer that
32:25
[Music]
32:44
[Music]
33:07
okay so it seems like answers have kind
33:10
of stabilized out there so let's go
33:11
ahead and grade things and in this case
33:13
the correct answer of the choices we
33:16
have comes out to be be that the log
33:19
base a of B involves taking one divided
33:21
by the log base B of a again let's think
33:25
this through both in terms of an example
33:26
and then in terms of a more poofy
33:29
systematic reason why it should be true
33:32
so if we're swapping our bases let's
33:37
just start off with our good old friend
33:40
log base 10 and let's plug in a nice
33:43
power of 10 like a thousand it's
33:47
counting the number of zeros we get
33:48
three so let's try swapping the bases
33:51
and see what this should mean log base a
33:54
thousand of 10 okay well what is this
33:58
asking right
34:00
maybe you can't think of drawing the
34:01
little triangle saying something like we
34:04
know a thousand to something is equal to
34:07
ten a thousand to the what equals ten
34:11
well if 10 cubed is a thousand that is
34:17
the same thing as saying ten is equal to
34:20
a thousand raised to the one-third doing
34:24
the inverse here involves the
34:27
multiplicative inverse of the exponent
34:29
and the way that pans out is that it
34:31
looks like 1 divided by 3 and that 3
34:35
corresponds to the log base 10 of a
34:37
thousand it's 1 divided by the log base
34:42
10 of a thousand so more generally you
34:47
might guess based on the single example
34:49
that when we swap the base with what's
34:52
on the inside
34:52
it corresponding it corresponds to
34:54
taking 1 divided by what's on the
34:56
outside there and again you can think
34:59
this through in terms of looking at the
35:01
corresponding exponential rule now what
35:03
happened to my lovely little log and
35:06
Exponential's wonderful so again let's
35:11
hide where some other thing
35:12
and the other properties that we'll get
35:14
to here and I'll keep it in the same
35:19
order I had it before here I was
35:21
thinking that having a pre-written could
35:22
keep me a little bit cleaner than usual
35:23
but maybe it just involves playing this
35:26
weird game of paper cutting shuffling
35:29
around and so what we just found log
35:31
base B of a if you swap those it's the
35:34
same as dividing by one what this
35:35
corresponds to often exponential land is
35:37
if you take B to some power and say that
35:41
that equals a that's the same statement
35:43
as saying that a to the inverse of that
35:45
power equals B again it's kind of
35:48
helpful to take a moment and think of
35:49
the logarithms as turning things inside
35:50
out the expression log base B of A is
35:54
playing the role of that X okay and the
35:57
expression log base a of B is playing
36:00
the role of whatever sits on top of the
36:02
a okay and then symmetrically the whole
36:06
expression B to the power X is playing
36:09
the role of the inside on the left it
36:11
plays the role of the a and the whole
36:13
expression a to the power of something
36:15
plays the role of what's sitting inside
36:17
the log base a there so you can see just
36:21
by plugging in some examples and by
36:23
corresponding it to the exponential
36:25
rules we can already think through three
36:28
different logarithm rules which if they
36:29
were just handed down as pieces of
36:31
algebra to be memorized you know you
36:34
could memorize them but it's very easy
36:35
for them to kind of slip out of your
36:37
head and it's also very easy to get
36:38
frustrated by the task at hand but you
36:41
might want to remind yourself that the
36:42
reason we care about these sorts of
36:44
things is understanding the rules of
36:46
logarithms helps us do math in contexts
36:48
where it's like a virus growing where
36:50
from one day to the next from one step
36:52
to the next things tend to grow
36:53
multiplicatively understanding the rules
36:55
of logarithms helps you to get a better
36:57
feel for that kind of stuff so before we
37:00
do a nice real-world example of what
37:02
that can look like let me just do one
37:03
more quiz question in this vein to ask
37:06
about properties of logarithms okay one
37:09
last one before we transition to a
37:11
little bit of a real-world example get
37:13
rid of what we had here
37:17
[Music]
37:18
and now which of the following is true
37:24
log of a plus B is the same as log of a
37:27
plus log of B log of a plus B is equal
37:30
to log of a times log of B log of a plus
37:33
B is equal to 1 divided by log of a plus
37:36
log of B or log of a plus B is equal to
37:40
1 divided by log of a times log of B or
37:42
none of the above
37:43
ah and now we don't have as much
37:47
consensus do we very interesting we've
37:49
got a horse race between two so I will
37:52
give you a moment to think this through
37:57
[Music]
38:08
[Music]
38:18
while people are answering actually I
38:20
have a little question for the audience
38:21
so you know I was just talking about how
38:24
we might think in terms of
38:25
multiplicative growth and that doesn't
38:28
just have to be powers of 10 we could
38:30
also do something like powers of 3 where
38:32
if you're going from 1 to 3 to 9 to 27
38:36
to 81 all of these we can say that the
38:40
log base 3 of these numbers just grows
38:44
in nice little steps so log base 3 of 1
38:48
3 to the what equals 1 they answer a 0
38:51
in general the log of 1 no matter the
38:53
base will be 0 log base 3 of 3 3 to the
38:55
1 equals 3 is 1
38:57
similarly log base 3 of 9 is 2 you might
39:03
wonder what my question is but it'll
39:05
help to draw all of these out and for my
39:08
own pleasure here let me just write out
39:10
one more log base 3 of 81 is 4 now
39:13
I've heard that ostensively if you ask a
39:16
child's let's say around like five or
39:18
six years old what number is halfway
39:20
between one and nine and you say what
39:24
number is halfway that their instincts
39:28
for how to answer are logarithmic
39:30
whereas our instincts tend to be more
39:32
linear so we often think 1 and 9 you've
39:35
got a bunch of evenly spaced numbers
39:37
between them 2 3 4 5 6 7 8 and if you go
39:40
right halfway in between I'll land on 5
39:42
but if you're thinking in terms of
39:43
multiplicative growth where to get from
39:46
1 to 9 it's not a matter of adding a
39:47
bunch of things but you're growing by a
39:49
certain amount you grow by a factor of 3
39:51
then you grow by another factor of 3
39:53
supposedly a kid's natural instinct
39:56
lines up with saying 3 and supposedly
39:58
this also lines up with if you have
40:00
anthropologists studying societies that
40:03
haven't developed counting systems and
40:05
writing in the same way that modern
40:07
societies have they'll enter 3 for this
40:09
so my question for the audience if any
40:11
of you watching right now have access to
40:13
a small child let's say in the range of
40:16
5 years old see if you can go ask them
40:19
what number is halfway between 1 and 9
40:21
and if you can let us know on Twitter
40:24
what the what the child says what their
40:26
actual answer is because I I don't know
40:29
why I'm
40:30
a little bit skeptical of whether that
40:31
actually pans out in practice I
40:34
understand this is not a super
40:35
scientific way to do it asking people
40:37
watching a YouTube livestream to survey
40:39
their own children and then tweet the
40:41
answer but for my own sake it would be
40:42
interesting to see some kind of
40:44
validation there back to our question
40:47
this is the first one that doesn't seem
40:48
to have a huge consensus in one
40:51
direction let's go ahead and grade it to
40:53
see what the answer turns out to be
40:57
great okay so 2,400 have you correctly
41:01
answered that it's none of the above
41:03
okay the log of a plus B doesn't satisfy
41:06
any of these nice properties and in
41:08
general unless we're going to be working
41:10
with certain kinds of approximations
41:13
especially when the natural log comes
41:14
into play we might talk about this next
41:15
time adding the inputs of a logarithm is
41:18
actually a very weird sensation it's a
41:19
very weird thing to do and to get a
41:22
sense of that weirdness plugging some
41:23
powers of 10 all right if I ask you log
41:28
of a plus B what you might start
41:31
thinking is okay let me just plug in
41:33
some examples like ten thousand and one
41:38
hundred and I ask myself if I do this
41:42
zero counting function of what's in that
41:44
input how many zeros are in it but it's
41:47
weird because when we add 10,000 100
41:50
well we're no longer at a clean power of
41:52
10 and okay that's fine you know often
41:55
you're taking logarithms of things that
41:56
aren't clean powers of 10 but it becomes
41:58
very strange to ask how you express this
42:01
in terms of log of 100 which was 2 + log
42:08
of 10,000 which was 4 because if you
42:14
look at log of 10100 it's asking 10 to
42:17
the what is equal to 10,000 100 you
42:21
might say I don't know it's gonna be a
42:23
little above 4 because it's kind of
42:24
close to 10,000 so the best you might
42:27
guess here is oh this is gonna be
42:30
something that's kind of like the log of
42:33
10,000 but that just feels like a
42:35
coincidence based on the two numbers
42:37
that we happen to put in there's not a
42:38
nice systematic reason coming there so
42:40
maybe you guess oh the numbers a and B
42:42
are very different
42:43
it's kind of close to whatever the
42:46
maximum of them is but it's very bizarre
42:49
and most importantly for the sake of the
42:52
quiz if you just look at the options
42:55
that it's giving you if you try this out
42:57
with any particular numbers you'll find
42:59
that none of those actually work so all
43:02
is good sometimes you get something that
43:04
looks like it's gonna be a nice property
43:05
but it doesn't end up being a nice
43:07
property and I also think that's
43:08
important rather than just finding
43:09
yourself only working with the various
43:12
you know log of a times B or log of X to
43:15
the power n these things that have a
43:16
nice rule sometimes you're out in the
43:18
mathematical wild you're working on some
43:20
problem you have a logarithm expression
43:22
and it's adding things in the input and
43:23
you want to be able to have familiarity
43:26
with the fact that that's kind of weird
43:27
that you're not going to be able to
43:28
simplify but if you you know if you
43:31
hadn't thought about that before you
43:32
might wonder oh is there just some
43:34
formula that I haven't seen before
43:36
so with all of that let me go ahead and
43:39
take a couple questions from the
43:40
audience before we transition to a
43:42
different sort of example
43:44
so it looks like uma Sherma asks can can
43:49
the bass be zero that's an interesting
43:51
question okay can the base of a
43:53
logarithm be zero well in terms of our
43:55
triangle we might think of that as
43:56
saying you know zero to some kind of
44:00
power X is equal to some other value Y
44:03
all right
44:04
this is something that we could write
44:06
either by saying zero to the x equals y
44:08
or we could write the same thing by
44:11
saying log base zero of y is equal to x
44:16
0 to the what equals x now the issue
44:19
here is that 0 to anything ends up being
44:21
zero right so if we're just going to be
44:25
thinking of log base zero of Y for any
44:30
other input Y you know you want to input
44:32
something like 1 or 2 or pi anything you
44:36
might want you're asking the question
44:38
zero to the what is equal to 1 or 2 or
44:43
PI or whatever number you might have
44:46
there and there's just not going to be
44:48
an answer so at best
44:50
you could try to say oh yes log of 0
44:52
it's a perfectly valid function it's
44:55
only defined on the input 0 but even
44:57
then you'd have trouble trying to
45:00
finagle what you want there because
45:01
saying 0 to the y equals 0 it's like
45:03
anything anything applies to it so your
45:07
arm is going to be twisted behind your
45:08
back however you want to make that work
45:09
and it corresponds to the fact that the
45:11
exponential function with base 0 is
45:15
entirely 0 it doesn't it doesn't map
45:18
numbers in a nice one-to-one fashion on
45:21
to each other so that's a great question
45:24
can you have a log base 0 now back to
45:26
the idea of where these things come up
45:28
in the real world one one example I kind
45:31
of like is the Richter scale for
45:32
earthquakes
45:33
so the Richter scale gives us a
45:37
quantification for how strong an
45:39
earthquake is okay and it can be
45:41
anything from very small numbers up to
45:43
very large numbers like I think the
45:45
largest earthquake ever measured and
45:47
this is just a chart that comes from
45:48
Wikipedia was a 9.5 okay and to
45:52
appreciate just how insane that is it's
45:55
worth
45:57
looking at the relationship between what
45:59
these numbers mean and then something
46:01
like the equivalent amount of TNT some
46:03
sort of measure of how much energy there
46:05
is in it and then what we can try to do
46:08
here is see if we can get an expression
46:09
for the Richter scale number in terms of
46:12
the amount of energy and why logarithms
46:15
will be a natural way to describe this
46:16
so the key to focus on is as we're
46:19
taking steps forward how much do things
46:21
increase so for example if we go from
46:24
two well in this case it doesn't show us
46:27
where three is so maybe we think of
46:28
taking a step from two up to four which
46:30
is kind of like taking two steps what
46:32
does that do in terms of the amount of
46:33
energy well it looks like it takes us
46:35
from one metric ton of TNT which is I
46:38
guess a large bomb from World War two
46:40
and it takes us up to a kiloton a
46:42
thousand times as much okay which is a
46:45
small a small atom bomb so just two
46:48
steps on the Richter scale going from an
46:50
earthquake of magnitude two to an
46:51
earthquake of magnitude four takes us
46:53
from large bomb from World War two up to
46:56
the nuclear age so that is noteworthy
47:00
and the first clean step that we get is
47:04
going from four to five at least in
47:06
terms of what this chart is nicely
47:07
showing us and evidently a single step
47:10
up from four to five corresponds to
47:12
going from 1 kiloton to 32 kilotons and
47:16
that was everything that was a 4.0
47:18
versus an earthquake there was a 5.0
47:20
it's easy to think yeah 4 and 5 those
47:22
are pretty similar numbers but evidently
47:24
in terms of TNT amounts that corresponds
47:26
to multiplying by 32 to get from one to
47:29
the next and going from 2 to 4 was
47:32
evidently multiplying by about a
47:34
thousand okay and the only reason that's
47:38
bigger is because here our chart wasn't
47:40
showing what 3 was so we were taking two
47:41
steps and you can verify for yourself
47:44
that if you take a step of 32 and then
47:47
you multiply by another 32 that's
47:49
actually pretty close to a thousand so
47:52
the idea that additive steps on the
47:54
Richter number correspond to
47:55
multiplicative steps in the TNT seems to
47:58
suggest that something logarithmic is at
47:59
play here and it's a little interesting
48:02
to just keep going here and say how how
48:05
much does this grow partly
48:08
because of the the world phenomena it's
48:10
describing yes not a huge surprise that
48:12
as we take another step
48:14
it's multiplying by about 32 again but
48:18
raining that in to our intuitions that's
48:21
the difference between 32 kilotons a
48:23
small atom bomb and then one Megaton
48:26
which we might think of as not small
48:28
atom bomb Nagasaki atom bomb which I
48:31
guess is 32 of the Nagasaki atom bombs
48:33
for one Megaton that is evidently the
48:36
magnitude of the double screen flat
48:39
earthquake in Nevada USA in 1994 I
48:41
didn't know what that was
48:42
thanks for capilla in terms of
48:44
frequencies by the way I also looked
48:47
these up evidently ones that are less
48:50
than two those happen all the time
48:51
there's like 8,000 of those per day but
48:53
as soon as we're in the realm of atom
48:55
bombs things like 3.5 and for those
48:58
evidently also happen quite frequently
49:00
somewhere on the earth there's around
49:01
134 of those happening somewhere every
49:04
day who knew but as we get even more
49:07
intense into this five and six range
49:09
which you know we're well above the atom
49:12
bomb scale
49:13
now we're only merely at around two per
49:16
day and you know I'm sure that a
49:18
geologist can come in and explain why we
49:20
all shouldn't be super worried about the
49:21
fact that there's two atom bomb
49:23
equivalent disruptions to the Earth's
49:25
crust happening every day but presumably
49:27
it's particularly rare for those to be
49:29
concentrated on some some spot like a
49:32
city where lots of people live now just
49:34
verifying that our thought that each
49:36
step involves a growth of 32 let's look
49:40
at what the step from 6 up to 7 looks
49:42
like and here it's giving us a lots more
49:44
examples in between maybe giving the
49:46
illusion that that's a bigger step that
49:47
it actually is and indeed that's the
49:49
difference between one Megaton and 32
49:51
megatons
49:52
so that's multiplying by 32 one of the
49:56
things I found most interesting on this
49:57
chart by the way was look at how far we
50:00
have to go before we get to the largest
50:03
nuclear weapon that's ever actually been
50:06
tested this was height of the Cold War
50:09
the Tsar bomb that was 50 megatons and I
50:13
believe they actually had original plans
50:15
to have a 100-megaton bomb but talked
50:17
themselves down from that 50 megatons
50:20
we're talking start off at that 30
50:22
to kilotons of the Nagasaki bomb
50:23
multiplied by 32 to get a Megaton
50:26
multiplied by another 32 rights we're
50:29
talking about a thousand times the
50:30
strength of the world war two ending
50:32
explosion and you're still not at the 50
50:35
megatons of what humanity is capable of
50:37
and that is evidently you know the Java
50:40
earthquake of Indonesia so 7.0 is not
50:43
just a little bit bigger than 6.0 it's a
50:45
lot bigger and the point here of course
50:47
is just that when you have a scale
50:49
giving you multiplicative increases it's
50:51
worth appreciating that what looked like
50:53
small steps can actually be huge steps
50:55
in terms of the energy implied or the
50:57
absolute values implied here so it I
51:01
mean when we're thinking about the fact
51:02
that there was ever a 9.5 that actually
51:04
seems absurd given that it's only in the
51:06
7.0 range that we're talking about the
51:08
largest thermonuclear weapon ever put
51:10
out and this is indicative of one area
51:12
where logarithms tempt to come about
51:14
it's um when humans want to create a
51:17
scale for something that accounts vary
51:19
hugely wide variance in how big things
51:21
can be so in the case of size of
51:24
earthquakes you can have things from
51:25
what happens just all the time around
51:28
the earth the size of a lady right and
51:30
in order to have that in a way that
51:31
you're not just writing a whole bunch of
51:33
different digits in your numbers for one
51:34
case and a whole bunch of different a
51:36
smaller number of digits for your number
51:38
in another case it's nice to take
51:40
logarithms and then just put that on a
51:42
single scale that basically puts
51:43
squishes those numbers between zero and
51:45
ten you see something very similar going
51:48
on with the decibel scale for music that
51:51
one actually works a little bit
51:52
differently where every time you take a
51:54
step up of ten decibels that corresponds
51:56
to multiplying by ten so rather than a
51:59
step of one multiplying by ten it's a
52:01
step of ten that multiplies by ten so
52:03
that kind of makes the math of it a
52:04
little bit screwy but the idea is the
52:06
same that if you're listening to a sound
52:08
that's 50 decibels for 60 decibels it's
52:11
a lot quieter in terms of the energy
52:13
being transmitted and going from you
52:15
know what would it be 60 to 70 or 70 to
52:17
80 those steps you know from 60 up to 80
52:21
that involves multiplying the amount of
52:22
energy per square area by a factor of
52:25
100 so every time you see a logarithmic
52:27
scale know in your mind that that means
52:30
whatever it's referring to under the
52:31
hood grows by a huge amount
52:34
this is again why we saw a lot of
52:35
logarithmic scales used to describe the
52:37
corona virus outbreak so how might you
52:40
describe a relationship like this where
52:42
every time you grow the Richter scale
52:44
number by one you're multiplying by 32
52:47
well we could think in terms of a log
52:50
with base 32 I could say if I take the
52:56
log of I'm just going to call R the
52:59
number for the Richter scale I might
53:01
think of this as log base 32 and that's
53:05
going to correspond to no no no I'm
53:07
doing this wrong no that's not the thing
53:09
that's logged we take the log base 32 of
53:12
the big number of the the T&P number
53:15
something that was like you know one
53:18
Giga ton or one Megaton it's 1 million
53:24
tons the log base 32 that should
53:29
correspond to the Richter scale number
53:31
but there might be some kind of offset
53:34
so we might say that there's some kind
53:36
of constant s that we're adding to this
53:39
Richter scale number and this expression
53:41
is exactly the same excuse me for going
53:44
off the bottom there this expression is
53:46
exactly the same as saying 32 to the
53:51
power of some offset times our Richter
53:53
scale number which is the same as taking
53:56
no 32 to that offset which itself is
54:01
just some big constant times 32 to the
54:05
Richter scale number so you might think
54:07
of this as just being some constant
54:09
times 32 to the power of the number you
54:12
see so this way of writing it really
54:15
emphasizes the exponential growth of it
54:17
that if this is what corresponds to the
54:20
TMT amount that you see as you increase
54:22
that are step by step you're multiplying
54:24
by 32 but another way of communicating
54:27
the exact same fact is to take the log
54:29
base 32 of whatever that amount is all
54:33
right now the next thing I want to talk
54:35
about is how we don't always have to
54:37
worry about how to compute logs of
54:40
different bases and it's a little weird
54:42
here that we were talking about log base
54:43
32 I referenced earlier how
54:45
mathematicians really like to have
54:47
with base e computer scientists really
54:50
like to have a log with base 2 and it
54:52
turns out for computational purposes or
54:56
for also thinking about how these things
54:57
grow if you have one log if you're able
55:00
to compute one type of log whether
55:01
that's base 10 base to base e you can
55:04
compute pretty much anything else that
55:05
you want okay now to get our intuitions
55:09
in that direction let's turn back to our
55:11
quiz and go to the next question and I
55:16
believe that this question is the most I
55:18
don't know this is this is a half way
55:20
reasonable question this should be nice
55:23
this is just gonna get us prepared to
55:25
translate from base to contexts to base
55:28
ten context and it's also a good
55:29
intuition for understanding powers of
55:32
two to have in general the relationship
55:35
that it has with powers of ten because
55:36
it's this lovely kind of coincidence of
55:39
nature that these two sort of well
55:43
you'll you'll see what I mean they play
55:44
nicely with each other so a question
55:46
asks given the fact that two to the
55:48
tenth is 1024 1024 which is
55:51
approximately 1000 okay so you can if
55:54
you're being a little bit loose with
55:56
your numbers and you're just making
55:56
approximations to to the tenth basically
55:59
a thousand which of the following is
56:01
closest to being true log base two of 10
56:04
is approximately zero point three log
56:07
base 2 of 10 is approximately sorry log
56:10
base 10 of 2 is approximately zero point
56:12
three log base 2 of 10 is approximately
56:15
one third or log base 10 of 2 is
56:17
approximately one third in which of
56:19
these is closest to being true based on
56:22
the fact that 2 to the 10th is
56:23
essentially a thousand I'll give you a
56:25
little moment for that
56:55
interesting that we've got kind of a
56:57
split on this one so I'm wondering if
57:01
they're gonna be numerically pretty
57:02
similar or if they're gonna be
57:04
conceptually similar or if there's even
57:06
a difference between those two
57:08
[Music]
57:09
so since answers keep rolling in I'm
57:11
gonna give this give us a little bit
57:13
more time so anyone at home watching
57:15
hopefully you already have a pencil and
57:17
paper out to be noodling through these
57:19
yourself that is the spirit of the
57:20
lectures that we're doing if you don't
57:22
now is the time to take out a pencil and
57:24
paper and see if you can think this one
57:27
through and write it out some of the
57:28
problems that we're gonna build to here
57:30
definitely will require pencil and paper
57:32
so now is as good a time as any and if
57:34
you're watching this in the future even
57:35
if you can't participate in the live
57:37
poll I really do think it's a lot of fun
57:39
to kind of throw your own hat into the
57:41
mix even if it's not going to contribute
57:43
to one of the numbers that you see
57:45
growing on the screen give you a little
57:49
bit more time here as the answers seem
57:51
to continue rolling in
57:58
[Music]
58:14
you
58:32
[Music]
58:53
okay so I'll go ahead and grade it now
58:56
and let's see how people did on this one
59:00
so the correct answer is B which is that
59:03
the log base 10 of 2 is approximately
59:06
0.3 and it looks like 1850 of you
59:09
correctly got that so congratulations
59:11
but the close contender not at all a
59:15
unanimous decision here
59:18
looks like it was d which is that the
59:20
log base 10 of 2 is around 1/3 so that's
59:22
good they're very numerically similar
59:24
right that it's either 0.3 or around 1/3
59:28
which is 0.33 3 3 repeating but the
59:30
question was asking which one is closest
59:32
to being true and let's see how we can
59:34
think about this so it points out that
59:37
you have a power of 2 which is a
59:40
thousand 24 awfully close to a power of
59:42
10 about 10 cubed and the question is
59:45
how we can leverage this to understand
59:46
something like log base 2 of 10 or log
59:51
base 10 of 2 as we saw earlier those are
59:52
just the reciprocals of each other so
59:54
what does this mean if log base 2 of 10
59:56
is equal to X that's the same thing as
59:58
saying 2 to the X is equal to 10 right
60:02
it's asking us 2 to the what equals 10
60:04
so what we have here is an expression 10
60:09
cubed is approximately equal to 2 to the
60:11
10 so what I might write out is we know
60:14
that 2 to the 10th instead of writing it
60:17
as a 10 I'm going to write that 10 as 2
60:20
to the X where X is the number such that
60:23
2 to the X is approximately is equal to
60:26
10 so if that cubed is the same as 2 to
60:30
the 10th this is I'll just write out the
60:33
full details the same as saying 2 to the
60:35
3x is equal to 2 to the 10th and
60:39
exponentiation is a nice one-to-one
60:41
function so it's okay to just say
60:44
whatever's going on in the input if the
60:46
outputs are the same the inputs must
60:47
also be the same you can't do that with
60:49
every function people seem to think you
60:50
can do that with any function but you
60:52
just can't and what that means is that X
60:56
is about X is about 10 thirds okay which
61:04
good so
61:05
place two of ten is about ten thirds so
61:09
if we looked at our answers though
61:10
that's not actually any of the options
61:13
we've got various things asking log base
61:15
two of ten being around 0.3 or one-third
61:18
so it looks like instead we should try
61:19
to reexpress this as log base 10 of 2
61:22
and well-enough what we saw earlier is
61:25
that log base 2 of 10 we could also say
61:28
log base 10 of 2 is just 1 over that
61:33
amount 1 over X and you can see this
61:35
pretty easily by writing 2 is equal to
61:37
10 to the 1 over X if we're asking 10 to
61:40
the what equals 2 the answer is 1 over
61:43
what we just got there so log base 10 of
61:47
2 is 1 divided by this amount which is
61:51
3/10 which is 0.3 great so this is kind
61:57
of a nice constant to think about
61:58
because there's this wonderful pattern
62:00
that happens when we're looking at
62:01
powers of two so if I ask what is the
62:05
log base two of a thousand like we just
62:09
saw it's approximately the case that 2
62:12
to the power 10 is equal to a thousand
62:15
and because we're doing things in logs
62:16
I'm just gonna be writing it in that way
62:17
a log two of a thousand is approximately
62:20
10
62:21
similarly log base 2 of a million well
62:27
let's see if we have to multiply two by
62:30
itself about ten times to get to a
62:32
thousand we should have to multiply by
62:33
itself around 20 times to get up to a
62:36
million
62:36
and indeed log base two of a million is
62:39
approximately 20 it's a little bit
62:42
smaller but this is kind of a nice
62:43
approximation to have in your mind and
62:45
then similarly you'll see why I'm
62:47
writing out this as a pattern in just a
62:49
moment if we wanted to go up to a
62:51
billion saying how many times do I have
62:54
to multiply two by itself to get to a
62:56
billion this is about 30 and any
62:59
computer scientist out there who's
63:01
thought about you know just how much is
63:03
a kilobyte or a megabyte or a gigabyte
63:05
they'll be familiar with the idea that
63:07
powers of two are nice and close to
63:10
these powers of ten or more specifically
63:12
powers of a thousand now what I want to
63:16
do is just write all of the same things
63:18
log base 10 not not approximately equal
63:22
to this is actually equal to 3 log base
63:24
10 of a thousand sequel the 3 log base
63:29
10 will you tell me what's log base 10
63:31
of a million it's counting the number of
63:33
zeros it ends up being about 6 and log
63:36
base 10 of a billion counting the number
63:39
of zeros it ends up being nine now the
63:44
reason I wanted to write all of this out
63:46
it just emphasizes an interesting
63:47
pattern here which is we're just growing
63:50
by these increments right as we go from
63:52
a thousand to a million to a billion
63:53
with log base 2 we're stepping up by
63:56
steps of 10 but when we're playing the
63:58
same game with 10 we're stepping up by
64:01
these increments of 3 so there's this
64:03
nice relationship and in fact for all of
64:05
them to go from log base 2 to log base
64:08
10 it seems like we're just multiplying
64:10
by 0.3 so 10 times 0.3 is 320 we scale
64:16
down by that same amount 30 we scale
64:18
down by that same amount okay now this
64:23
is an intuition worth remembering if you
64:25
have your numbers described with one
64:28
base it's basically the same as
64:30
describing them with another base but
64:32
there's some rescaling constant okay now
64:35
the next question is going to start
64:36
getting us at that direction but it's
64:37
going to be framed in a way that just
64:39
looks like a whole pile of algebra and
64:40
again I will encourage you to plug in
64:42
numbers if you want to to gain a little
64:44
intuition for it so as our third to last
64:47
question this will be a long lecture we
64:51
have which of the following is true even
64:53
just a whole pile of various possible
64:57
ways to combine log base C of B times
65:00
log base C of a does that equal log base
65:03
B log base B of A and rather than me
65:05
reading them out to you I'll just let
65:07
you look through them plug in some
65:09
numbers I'll give you I'll give you a
65:10
meaningful time on this one because it's
65:12
not it's not obvious unless you're
65:14
already familiar with logarithms and
65:16
it's worth thinking through a little bit
65:29
[Music]
65:31
we have an outstanding question from the
65:33
audience which is does the bar length of
65:36
the pole use some kind of log function
65:39
and healthily it looks like Ben eater
65:41
has gone ahead and directly answered in
65:44
the form of the code involved where the
65:46
chart max is math dot is that raising
65:50
two to the power of a ceiling of a log
65:53
base two of the maximum attempt count
65:55
which I think is to say unraveling if
65:59
you're looking at the maximum number I'm
66:01
not I'm not great at Vanna whiting this
66:03
thing if you look at the maximum number
66:04
in our poll it's asking what's the log
66:07
base two of that so as it crosses
66:10
different powers of two then that
66:11
rescales it and yes yes is the answer
66:15
what a fantastically apropos question
66:17
thank you Karen all right so answers are
66:21
still rolling in and I think like I said
66:24
I just want to give you some more time
66:25
to think this too because it looks like
66:28
a big pile of algebra plug in some
66:30
numbers to see what seems to work well
66:33
and see which answer fits
66:49
[Music]
67:07
[Music]
67:29
[Music]
67:44
okay so even if you are still thinking
67:46
about it I'm gonna go ahead and grade it
67:48
here and then start talking about why
67:50
it's true and then also why we should
67:52
care why this is an operation that
67:54
actually tells us something so the
67:56
correct answer which it looks like
67:58
around 1700 of you got congratulations
68:01
is log base C of B times log base B of A
68:05
is equal to log base C of a great now
68:11
that's just a big ol pile of things why
68:12
would that be true how do we think about
68:14
it now
68:15
phase one like I said me we might plug
68:16
in an example but let's try to actually
68:18
think about why the example holds so let
68:21
me pull out we've got this as one more
68:25
of the log rules this is just repeating
68:29
the end look what the correct answer
68:30
turned out to be where we've got an
68:33
expression for hug base C of B log base
68:36
B of A and this ends up being log base C
68:38
of a it's gotten rid of the bees which
68:41
is kind of interesting so some examples
68:44
you might plug in here would be things
68:45
like let's use a different color that's
68:47
huge green instead of C I'm gonna go
68:53
ahead and plug in ten log base 10 of 100
68:57
which is kind of asking how many times
68:59
does 10 go into a not 100 in a
69:01
multiplicative sense how many times do I
69:03
multiply 10 by itself to get to 100
69:05
where the answer is 2 and then log of
69:08
100 of let's plug in another power of 10
69:12
it'll be nice if it's also a power of
69:14
100 so I'll do a million so this one is
69:19
asking ten to a hundred to the what
69:21
equals a million how many times do I
69:23
multiply a hundred by itself to get to a
69:25
million how many times does a hundred go
69:27
into a million phrasing the same thing
69:29
ten different ways now the claim is that
69:31
this is the same thing as taking log
69:33
base 10 of a million that if I ask how
69:40
many times does 10 go into 100 and how
69:42
many times does 100 go into a million
69:44
multiplying those should give me the
69:46
answer to how many times 10 goes into a
69:49
million now just checking the numbers
69:51
this certainly works
69:55
10 goes into a hundred two times 100
69:59
goes into a million three times in a
70:02
multiplicative sense in that 100 cubed
70:04
is equal to a million and indeed how
70:08
many times does 10 go into a million
70:10
well 6 now we can think of this property
70:13
in terms of the corresponding exponent
70:15
rule which is going to look a little bit
70:17
stranger but it's actually just saying
70:19
entirely the same thing so here we're if
70:23
we have a base of C in a base of B and
70:26
we're trying to relate those to each
70:27
other the whole statement is equal to
70:30
saying that um suppose the B to the X is
70:33
equal to a I've got some number B you
70:35
raise it to some number X and a equals a
70:37
suppose it's also the case that C to the
70:39
y equals B those two together are the
70:42
same as saying C to the X y equals a now
70:45
that's kind of a mouthful to say out
70:46
loud but if you plug in some numbers to
70:49
translate what all of that is really
70:50
saying in the context of the example we
70:53
just did saying if you can write a
70:55
hundred is 10 squared and if you can
70:57
write a million as 100 cubed well that
71:00
lets you write a million in terms of a
71:01
power of 10 yeah so sort of asking this
71:05
question how many times does one number
71:08
going to another but letting you layer
71:10
it on top of each other now if we
71:13
rearrange that expression we get what is
71:15
probably the second most important of
71:17
all of our log rules the most important
71:19
is this top one that when you multiply
71:20
the inputs you add the outputs but the
71:23
second most important which is known as
71:25
the change of base formula lets us write
71:29
that if you want the log base B of some
71:32
value a then for what did we see you
71:35
want it doesn't actually matter what log
71:37
you have in your pocket if you use that
71:39
other log and you take the log C of a
71:41
divided by the log C of B that gives you
71:44
log base B of a ok so just as an example
71:47
here what this would look like is let's
71:50
say I wanted to be able to compute log
71:52
base 100 of things I just really want to
71:55
it's not a button on my calculator but I
71:57
would love to be able to do it and I
72:00
want to do it in with an input like a
72:03
million well even if I don't have the
72:05
log base 100 button on my calculate
72:07
what I can do is say I'll use the log
72:10
base 10 button and evaluate what's on
72:14
the inside here which at least
72:16
positionally it's kind of above the 100
72:18
it has a higher altitude as we write it
72:21
so this can line up with the notation a
72:24
little bit that it sits on the numerator
72:26
and on the bottom I use the log base 10
72:30
button that's in my calculator on the
72:32
base on the 100 and then I can evaluate
72:35
both of those and will give me the
72:37
answer in this case it gets you 6
72:38
divided by 2 which will be 3 and if we
72:42
really just think through what this is
72:44
saying I know I've said it many
72:45
different times but it's a it's a
72:47
convoluted enough way to write things
72:48
but an intuitive enough fact that I
72:51
think just coming at it from a bunch of
72:52
different angles can be important
72:54
because like I said this is probably the
72:56
second most important log rule we're
72:58
asking how many times does 100 go into a
73:01
million in a multiplicative sense how
73:04
many times we multiply by itself but
73:06
division is asking that same question in
73:08
an additive sense if I say how many
73:10
times does the log of 100 go into the
73:12
log of a million that's what division
73:15
means it's saying how many times do I
73:16
add this bottom number to itself to get
73:18
to the top but anything additive in the
73:21
logarithm realm is the same as anything
73:23
multiplicative in terms of what's inside
73:25
the parentheses so both of the left-hand
73:27
side and the right-hand side are just
73:29
saying how many times does 100 go into a
73:31
million but going about that in
73:33
different ways so this is extremely nice
73:35
because it actually lets us compute
73:37
things next time we're going to talk all
73:39
about the natural logarithm which is log
73:41
base E often written Ln and turns out
73:45
this is much easier to compute there's
73:47
nice math behind it such that if you
73:49
want to come up with an algorithm that
73:51
your calculator can use it's actually a
73:53
lot easier to think of log base E of
73:55
numbers so anytime that you need to go
73:57
to some kind of calculator I don't know
73:59
let's say we popped over to something
74:01
like desmos always happy to have as a
74:03
friend and you wanted to compute log
74:06
base 10 of some number you know let's
74:09
say we're doing a log base 10 of 57 and
74:12
it looks like that's you know it should
74:14
make sense is between 1 and 2 because 57
74:17
was between 10 and 1
74:18
hundred what's going on under the hood
74:20
how is it actually figuring this out
74:22
it's gonna be use somewhere in there a
74:24
change of base formula which is going to
74:26
be that the natural log of 57 divided by
74:30
the natural log of 10 is the same thing
74:32
these are two different ways of writing
74:34
it so if you know one logarithm you know
74:36
all of the logarithms yeah and well
74:41
let's just use that fact to answer one
74:42
more of our quiz questions and this will
74:44
be the second to last quiz question so
74:46
thank you all for sticking it through I
74:48
think we will be I think you'll be
74:52
pleased by the last question because the
74:54
last question will actually be like a
74:55
fun problem solving puzzling thing and
74:57
it'll involve a lot of what we've used
74:59
up to this point kind of a culminating
75:01
thing so before that though just to make
75:04
sure that we've got the instincts of
75:06
change of base down what do we have use
75:10
the approximation that log base two of
75:13
ten is around ten thirds so using that
75:16
approximation which of the following is
75:18
approximately true log base two of X is
75:21
about ten thirds of log base 10 of X log
75:25
base two of X is log base ten of ten
75:28
thirds of X log base two of X is log
75:31
base 10 of X to the power 10 thirds or
75:34
log base two of X is ten thirds times
75:37
the log base 10 of X and finally none of
75:39
the above so we'll give you a moment to
75:42
think about that you might want to think
75:43
to the chart that we were drawing
75:45
earlier and thinking about log base two
75:47
and log base ten as we're looking at
75:49
powers of 1000 and how each of those
75:52
grows that can leverage some of the
75:53
intuition but I'll let you think about
75:55
it stop talking
76:09
you
76:24
while answers are rolling in it looks
76:26
like a number of people have been asking
76:28
on Twitter about basically how complex
76:31
numbers play into this so you know we've
76:33
got general asking what if the base is
76:36
imaginary
76:37
we've got Kalkan asking what about
76:39
complex bases since e to the X is
76:42
walking around the spiral wouldn't it
76:43
hit every complex number
76:44
well it won't hit every complex number
76:46
but your instinct that it's hitting
76:48
multiple things is pretty spot-on does
76:51
it make sense to talk about Logs with
76:52
imaginary numbers by Mitya so to all of
76:55
these questions it's actually a very
76:59
nuanced question this the short answer
77:01
is yes complex logarithms absolutely
77:03
exist but each one of them has multiple
77:05
different outputs so a good a good
77:07
analogy here is how if we have the
77:10
square root function if I ask something
77:12
like the square root of five you know we
77:15
have the convention that you always do
77:17
the positive amount but that doesn't
77:18
quite feel honest it feels like the
77:20
right answer is to specify that there's
77:22
two different outputs for the square
77:24
root function two solutions to x squared
77:27
equals 25 and this is actually true in
77:31
complex numbers as well one of the
77:33
things we talked about in a complex
77:34
number lecture was that you can take the
77:36
square root of I and you actually get
77:38
root 2 over 2 plus root 2 over 2 I but
77:43
that there's two solutions you can do
77:45
plus or minus this value and so you
77:48
could say that the square root function
77:49
isn't a function but it's a multivalued
77:52
function it always has two different
77:54
outputs
77:54
now something funky happens when we have
77:56
exponents at play so if you're just like
77:59
someone who hasn't seen this stuff don't
78:01
worry we talked about it in previous
78:02
lectures I'm obviously jumping around on
78:04
the complexity level a lot here where if
78:06
in lecture 5 I'm talking about like
78:09
Euler's formula with complex numbers and
78:11
compound interest and like how that
78:12
plays into physics and then lecture 6
78:14
we're back to the basics of logarithms I
78:15
acknowledge that might be a little bit
78:18
jarring to potential audience members
78:20
but just continuing on with the answer
78:22
if you note that e to the 2 pi times you
78:29
know some number times I
78:32
this basically walks you around a circle
78:35
so that the output will walk around a
78:39
circle and just keep repeating as n goes
78:41
from 0 up to 1 it'll walk around one
78:44
cycle and end up back where you started
78:45
as n goes from 1 to 2 you'll end up back
78:48
where you started so for example e to
78:50
the 0 sits here e to the I pi is at
78:55
negative 1 but e to the 2 pi is also at
78:59
1 same with e excuse me same with e to
79:07
the 4 pi i that also equals 1 so in
79:12
general if you wanted to ask something
79:13
like what is the log base e of 1 you
79:19
know the one hand we want to say the log
79:21
of log base anything of 1 should be 0
79:23
because anything to the power 0 equals
79:25
that 1 but if we're letting complex
79:27
numbers into the mix you would have to
79:29
honestly say well 2 pi is another pretty
79:32
good answer to this question because e
79:34
to the 2 pi also equals 1 and same with
79:37
4 pi and you could even go in the
79:39
negative direction and in general n
79:41
times pi sometimes 2 times I kind of
79:45
wrote that in a weird order for any
79:47
integer in feels like a valid answer to
79:53
this question so there's a couple of
79:55
ways that you can deal with that in math
79:57
and before we get back to our usual
79:59
lesson just on logarithm rules this is
80:01
interesting enough I kind of want to
80:02
pull it up let's see what if we have
80:05
complex logarithm wonderful let's zoom
80:10
out a little bit great so there's this
80:13
notion of what's called a Riemann
80:14
surface that's basically trying to
80:16
capture the idea that you have a
80:17
function with multiple outputs and
80:18
intuitively maybe you can understand
80:20
what it's getting at where the input
80:21
would be something on the XY plane and
80:23
the output there's just many different
80:24
output sitting there so when you have a
80:26
complex logarithm you have to account
80:29
for that but it's used it's actually a
80:30
very useful idea to do but it takes a
80:34
lot more nuance than than you might
80:35
expect
80:36
so with all of that hopefully that
80:39
helped at least partially answer some of
80:42
people's questions seem same by the way
80:44
with
80:45
rhythms with based negative numbers
80:46
because if you're asking like negative
80:48
one to the X and really noodling on what
80:50
that means it's it gets you into the
80:54
realm of complex numbers so you have to
80:56
deal with the same multiple output idea
80:58
now answers are rolling in more slowly
81:02
so this seems like a fine time to grade
81:04
things and the answer turns out to be
81:07
you rescale it if you want to go from
81:10
log base two of something to log base 10
81:12
of something it involves rescaling and
81:15
one way you could think about this is
81:16
with the change of base formula so let's
81:19
say we have let's get rid of our key
81:22
steps let's say you have log base 2 of X
81:29
but we want to write it in terms of log
81:31
base 10 we can write that as log base 10
81:34
of X divided by log base 10 of 2 saying
81:40
how many times 2 to go into X in a
81:42
multiplicative sense is the same as
81:44
saying how many times does the log of 2
81:45
go into the log of X in an additive
81:47
sense yeah and well what is 1 divided by
81:52
the log base 10 of 2 so we're going to
81:55
keep our log base 10 of X out here and
81:59
from what we found earlier we found that
82:02
log base 10 of 2 was approximately 3
82:06
tenths was approximately 0.3 so when we
82:08
divide by it that should give us 10
82:11
thirds okay and this lines up with what
82:14
we were looking at a little bit earlier
82:15
with powers of 1,000 let's see where was
82:18
it where was it great so when we're
82:23
converting from log base 10 of something
82:25
up to log base 2 you know here we were
82:27
thinking of multiplying the top by 0.3
82:29
to get to the bottom but you could also
82:30
think of multiplying by 10 thirds to get
82:32
to the top anytime you have log base 10
82:34
of some number you just rescale it and
82:36
you have log base 2 of that number and
82:38
the rescaling constant comes from the
82:41
log base 2 of 10 or vice versa log base
82:43
10 of 2 so that's change of base like I
82:46
said it's very important it´ll X you put
82:48
everything into a nice universal
82:49
language and that should be all of the
82:52
hint that you need for the last question
82:54
which is a challenge question on this
82:55
one so if anyone's been watching
82:57
and they've been like I know logarithms
82:58
I've got this completely down let me
83:03
pull up the last question which came up
83:04
on let's see it's not the AMC but it's
83:07
whatever the predecessor to the AMC was
83:09
I believe so it's you know it's gonna
83:12
involve a little cleverness and
83:13
manipulation we've got this long
83:16
sequence of fractions okay you take one
83:19
divided by the log base two of a hundred
83:21
factorial and remember hundred factorial
83:23
is 100 times 99 times 98 on the way all
83:27
the way down to one so one divided by
83:29
the log base two of that plus one
83:31
divided by the log base three of 100
83:34
factorial plus one divided by the log
83:36
base four of 100 factorial on and on and
83:38
on up until 1 divided by the log base
83:41
100 of a hundred factorial so this looks
83:44
rather intimidating right certainly
83:47
adding fractions is never fun adding a
83:49
hundred fractions seems even worse
83:51
dealing with a bunch of logs of
83:53
different bases seems like a pain and
83:55
the factorial is playing into this so
83:58
I'm just gonna give you you know given
84:00
that this is the wind down time I'm
84:02
going to give you two or three minutes
84:04
to start thinking about this and if you
84:06
don't get it it's totally fine we're
84:08
gonna walk through what the answer is
84:09
but I'm just gonna let people think
84:12
about this final challenge question
84:14
before we call it a day
84:17
you
84:38
[Music]
85:07
[Music]
85:13
you
85:15
[Music]
85:29
you
85:44
you
86:03
[Music]
86:25
so can I give you a little bit more time
86:27
on this one because it's definitely it's
86:30
definitely fun to work out and I think
86:33
if you know how to start then great but
86:35
if you don't know how to start just
86:37
letting yourself kind of work with a
86:38
couple of the different rules that we've
86:40
worked out before the change of base
86:43
comes into play if you'd like to use
86:45
that and just kind of keep manipulating
86:48
and if it feels like you're getting
86:49
something that's a little too messy see
86:51
if you come at it from a different angle
86:52
and because answers are rolling in a
86:57
little bit more slowly now what I'm
86:59
going to do is just start to describe
87:01
the explanation here and then come back
87:03
to grade it in just a moment here so the
87:07
expression that we have looks like I
87:10
just started writing it out while you
87:12
guys were working on it one divided by
87:14
log base two of 100 factorial plus one
87:17
divided by log base three of 100
87:18
factorial and before you even start the
87:20
fact the thing on the inside of the log
87:22
involves a big product of stuff should
87:24
actually feel good because logs like to
87:26
take in things that look like products
87:28
because of the most important property
87:29
that we have which is the idea that it
87:32
turns products into addition okay
87:35
so right away you know that you're
87:37
probably going to use that second thing
87:38
you notice is that it's uncomfortable to
87:40
have all of these different bases log
87:42
base two of something log base three of
87:43
something log base a hundred of
87:44
something so translating it all into the
87:47
common language should be helpful how
87:49
this change of base work well asking how
87:51
many times two to go into a hundred
87:53
factorial is the same as asking how many
87:56
times does the log of two divide into
87:58
the log of 100 factorial it doesn't even
88:01
matter what logarithm you use this log
88:02
could be base ten base e base two
88:04
doesn't matter this change of base
88:06
formula still holds now what that means
88:08
for our expression is instead of taking
88:10
one divided by log base two of a hundred
88:12
factorial I could write that as log of 2
88:18
divided by so let me draw a little
88:20
dividing line between what I'm doing
88:22
here log of 2 divided by log of a
88:26
hundred factorial okay and then
88:29
similarly log of 3
88:33
divided by
88:34
of 100 factorial so all I'm doing here
88:38
is taking the reciprocal so instead of
88:42
taking log of 100 factorial over log of
88:43
2 I've inverted it log of 2 over 100
88:46
factorial because that's what this
88:47
reciprocal is doing so with that as the
88:50
beginning I'm going to go ahead and just
88:51
grade this lock in the answers and see
88:55
how see how everyone's doing so it looks
88:59
like awesome around 1796 we always we
89:04
always draw a little bit north of
89:07
Ramanujan's number around 1,800 of you
89:10
answered that it's 1 which is correct
89:12
congratulations 70 of you answered 100
89:16
which we can maybe see where where that
89:18
our error would have come from those of
89:20
you that answered 0 that's interesting
89:22
it would imply that somehow you have
89:24
cancellation at play because these are
89:27
all positive numbers so thinking one of
89:28
them was negative so you could probably
89:30
gut-check against the idea that it would
89:31
be 0 41 of you want me to come up with a
89:34
number fun numerical fact about 69 but I
89:36
won't 28th of you said 2 and I think
89:39
yeah I think that's I think the
89:43
difference between 1 and 100 maybe would
89:45
be interesting to try to analyze but if
89:47
we go back to our answer I realize it
89:50
may be a little bit confused rewriting
89:51
our expression up on the top here as we
89:55
add all of these things and we can
89:57
continue up until log of 100 all divided
90:03
by log of 100 factorial the key is that
90:08
now all of the denominators are the same
90:10
so we can add the numerators and the
90:13
more plus sign is just going to equal
90:15
something that top is going to look like
90:18
log of 2 plus log of 3 plus on and on up
90:26
to log of 100 all divided by the log of
90:32
100 factorial and it seems like the only
90:34
little contention among those answering
90:36
the question is whether this should
90:38
simplify to be 1 or should it simplify
90:40
it to be a hundred and really the way we
90:44
can think about this is to just break
90:46
down that bottom part
90:47
in terms of what a factorial means I'll
90:50
go ahead and do this on a bottom part
90:51
since I didn't manage my paper real
90:53
estate very effectively here the log of
90:56
100 times 99 times 98 on and on times 2
91:03
times 1 is the same as adding all of
91:05
them and you can probably see this
91:08
mostly cancels out with the top the only
91:10
question you might have though is that
91:11
in the factorial can we keep multiplying
91:14
down until we get that one which doesn't
91:17
really make a difference but before you
91:19
think about it too much you might wonder
91:20
hang on you know on the bottom I'm
91:23
multiplying everything and I'm adding
91:25
this log of 1 but on the top from what
91:28
we found before I never saw that log of
91:29
1 does that make any difference and the
91:31
answer is no because log of 1 is saying
91:34
10 to the power of what equals 1 and the
91:36
answer is 0 so in fact taking the log of
91:40
100 factorial is the same as adding the
91:42
logs of all the numbers from 2 up to 100
91:44
and it simplifies down to 1 so to those
91:47
of you who got it congratulations to
91:49
those of you who feel like you maybe
91:50
have better intuitions for change of
91:52
base formula for the fact that
91:53
logarithms turn multiplication into
91:55
addition that's my hope in the next time
91:58
with this as a foundation of logarithms
92:00
that can be pointed back to what I would
92:02
love to talk about is what's known as
92:03
the natural logarithm log base E and try
92:06
to give an instinct for why that's
92:08
something that we care about ok why is
92:10
it that mathematicians just seem to
92:12
really love the number E sitting in the
92:14
base of their logarithms what does that
92:15
buy for them why does it show up in
92:17
nature so that will happen on Friday at
92:20
the same time as this lecture and I look
92:22
forward to seeing everyone there
92:24
[Music]
92:37
[Music]
92:42
[Music]
92:48
you
92:50
[Music]