# Definition:General Logarithm/Positive Real

## Definition

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.

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

### Common Logarithms

Logarithms base $10$ are often referred to as common logarithms.

### Binary Logarithms

Logarithms base $2$ are becoming increasingly important in computer science.

They are often referred to as binary logarithms.

A notation which is starting to take hold for the binary logarithm of $x$ is $\lg x$.

## 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.

## 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 $x$

## Also see

• Results about logarithms can be found here.