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: \[a^{\log_a b} = b\quad\text{and}\quad \log_a (a^b) = b.\]