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:

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

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 $x$

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 $a = e$

• Results about logarithms can be found here.

Linguistic Note

The word logarithm comes from the Ancient Greek λόγος (lógos), meaning word or reason, and ἀριθμός (arithmós), meaning number.

Sources

• 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $2$: Functions, Limits and Continuity: The Elementary Functions: $6$