logarithm

The logarithm is the inverse operation of exponentiation.

Formally,

\begin{aligned} y & = b^{x} \\ x & = \log_{b} (y) \end{aligned}

for a given base $b$ .

Graphically,

Rule	Formula	Pneumonic
Logarithm of Power	$\log_{b} (b^{x}) = x$	"reduce to exponent"
Power of Logarithm	$b^{\log_{b} (x)} = x$	"reduce to argument"
Power of Argument	$\log_{b} (x^{n}) = n \log_{b} (x)$	"exponent to coefficient"
Product of Arguments	$\log_{b} (x y) = \log_{b} (x) + \log_{b} (y)$	"split and add"
Quotient of Arguments	$\log_{b} (\frac{x}{y}) = \log_{b} (x) - \log_{b} (y)$	"split and subtract"
Change of Base	$\log_{b} (x) = \frac{\log_{a} (x)}{\log_{a} (b)}$	"divide by base as argument"

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