# Definition:General Logarithm

## Definition

### Positive Real Numbers

Let $x \in \R_{>0}$ be a strictly positive real number.

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

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

$\log_a x := y \in \R: a^y = x$

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

### Complex Numbers

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.

## Base of Logarithm

Let $\log_a$ denote the logarithm function on whatever domain: $\R$ or $\C$.

The constant $a$ is known as the base of the logarithm.

## Examples

### General Logarithm: $\log_\pi \pi$

The logarithm base $\pi$ of $\pi$ is:

$\log_\pi \pi = 1$

### General Logarithm: $\log_b 1$

$\log_b 1 = 0$

### General Logarithm: $\log_b \left({-1}\right)$

$\log_b \left({-1}\right)$ is undefined in the real number line.

## Also see

• Results about logarithms can be found here.