Suppose $b \in (0, \infty), b \ne 1$.

Definition: The logarithm function base b. $log_b : (0, \infty) \rightarrow R$ , the logarithm function with the base $b$ is defined as the inverse function for the exponential function base b for real numbers.

$log_b(y) = x$ if and only if $b^x = y$.