# Definition:Logarithm

## Natural Logarithm

### Positive Real Numbers

Let $x \in \R$ be a real number such that $x > 0$.

The **(natural) logarithm** of $x$ is defined as:

- $\displaystyle \ln x := \int_1^x \frac {\d t} t$

### Complex Numbers

Let $z = r e^{i \theta}$ be a complex number expressed in exponential form such that $z \ne 0$.

The **complex natural logarithm** of $z \in \C_{\ne 0}$ is the multifunction defined as:

- $\map \ln z := \set {\map \ln r + i \paren {\theta + 2 k \pi}: k \in \Z}$

where $\map \ln r$ is the natural logarithm of the (strictly) positive real number $r$.

## General Logarithm

### 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.

## Historical Note

The initial invention of the logarithm was by John Napier, who used a base $\dfrac {9 \, 999 \, 999} {10 \, 000 \, 000}$.

William Oughtred, who invented the slide rule, added a table of natural logarithms to his English translation of Napier's book.

In $1685$, John Wallis established that logarithms can be considered as exponents.

In $1694$, Johann Bernoulli made the same discovery.

Logarithms in their modern form are as a result of the original work done by Leonhard Paul Euler.

Euler was the first one to create a consistent theory of the logarithm of a negative number and of an imaginary number.

He also discovered that the logarithm function has an infinite number of values for a given argument.

## Sources

- 2008: Ian Stewart:
*Taming the Infinite*... (previous) ... (next): Chapter $5$: Eternal Triangles: Trigonometry