Logarithmic Function
- (mathematics) Any function in which an independent variable appears in the form of a logarithm; they are the inverse functions of exponential functions
\[f(a)+f(b)=f(a.b)\]
\[nf(a)=f(a^n)\]
\[nf(1)=f(1^n)=f(1)\]
\(nf(1)-f(1)=0\)\(\Implicate\)
\[f(x)=\log_ax\]
\[\log’_a(x)=\lim_{ \epsilon\to 0} \frac {\log_a(x+\epsilon)-\log_a(x)}{\epsilon}\]
\[\log’_a(x)=\lim_{ \epsilon\to 0} \log_a[\frac {x+\epsilon}{x}]^\frac 1\epsilon\]
\[\log’_a(x)=\log_a\{\lim_{ \epsilon\to 0} [\frac {x+\epsilon}{x}]^\frac 1\epsilon\}\]
\[\lim_{ \epsilon\to 0} [\frac {x+\epsilon}{x}]^{\frac 1\epsilon}=\lim_{ \epsilon\to 0} [1+\frac {\epsilon}{x}]^{\frac 1\epsilon}\]
\[\lim_{ \epsilon\to 0} [\frac {x+\epsilon}{x}]^{\frac 1\epsilon}=\lim_{ \epsilon\to 0} \{[1+\frac {\epsilon}{x}]^{\frac x\epsilon}\}^\frac1x\]