Definition:General Logarithm/Complex

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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:

$\log_a z := \set {y \in \C: a^y = z}$

where $a^y = e^{y \ln a}$ 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 $x$


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

  • Results about logarithms can be found here.