# Definition:General Logarithm/Complex

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## 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**.

## Sources

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