# Logarithm laws

The logarithm laws are the rules by which logarithms may be combined, and are derived from the definition of the logarithm and the index laws.

The basic rules are: \begin{align*} \log_a a &= 1 \\ \log_a (xy) & = \log_a x + \log_a y \\ \log_a (x^n) &= n \log_a x \\ \end{align*} From these we can derive other important rules: \begin{align*} \log_a 1 &= 0 \\ \log_a (x/y) &= \log_a x - \log_a y \\ \log_a (1/x) &= - \log_a x \\ \log_a \sqrt[n]{\vphantom{()}x} &= \tfrac{1}{n} \log_a x \end{align*}

We can change the base of a logarithm: $\log_a x = \frac{\log_b x } {\log_b a }$

And from the definition of logarithm, it is the inverse of exponentiation: $a^{\log_a b} = b\quad\text{and}\quad \log_a (a^b) = b.$