logarithm

Logarithm

For any positive numbers $M,N$ and positive base $b\ne 1$:
A logarithm tells us what exponent we need to raise the base to get a number:
If $b^y=x$ then ${\log }_b(x)=y$
Written as ${\log }_b(x)$ or $\ln (x)$ for natural log (base $e$)

Basic Properties

${\log }_b(M\cdot N)={\log }_b(M)+{\log }_b(N)$
${\log }_b(\frac{M}{N})={\log }_b(M)-{\log }_b(N)$
${\log }_b(M^n)=n\cdot {\log }_b(M)$
${\log }_b(b)=1$
${\log }_b(1)=0$
${\log }_b(b^n)=n$
${\log }_b(x)=\frac{\ln (x)}{\ln (b)}$ (change of basis)