# Definition:General Logarithm/Positive Real

## Contents

## 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

- Definition:Real Natural Logarithm: when $a = e$

- Results about
**logarithms**can be found here.

## Sources

- 1968: Murray R. Spiegel:
*Mathematical Handbook of Formulas and Tables*... (previous) ... (next): $\S 7$: Logarithms and Antilogarithms - 1997: Donald E. Knuth:
*The Art of Computer Programming: Volume 1: Fundamental Algorithms*(3rd ed.) ... (previous) ... (next): $\S 1.2.2$: Numbers, Powers, and Logarithms: $(9)$