Definition:General Logarithm/Complex

Definition
Let $z \in \C_{\ne 0}$ be a non-zero complex number.

Let $a \in \R_{>0}$ be a strictly positive real number such that $a \ne 1$.

The logarithm to the base $a$ of $z$ is defined as:


 * $\log_a z := \left\{{y \in \C: a^y = z}\right\}$

where $a^y = e^{a \ln y}$ as defined in Powers of Complex Numbers.

The act of performing the $\log_a$ function is colloquially known as taking logs.

Also known as
The logarithm to the base $a$ of $x$ is usually voiced in the abbreviated form:
 * log base $a$ of $x$

or
 * log $a$ of $x$

When $a = 2$, a notation which is starting to take hold for $\log_2 x$ is $\lg x$. This concept is becoming increasingly important in computer science.

Also see

 * Definition:Complex Natural Logarithm: when $a = e$