# Definition:Natural Logarithm

## Contents

## Definition

### 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 $r$.

## Notation

The notation for the natural logarithm function is misleadingly inconsistent throughout the literature. It is written variously as:

- $\ln z$
- $\log z$
- $\log_e z$

The first of these is commonly encountered, and is the preferred form on $\mathsf{Pr} \infty \mathsf{fWiki}$. However, many who consider themselves serious mathematicians believe this notation to be unsophisticated.

The second is ambiguous (it doesn't tell you which base it is the logarithm of).

While the third option is more verbose than the others, there is no confusion about exactly what is meant.

## Examples

### Natural Logarithm: $\ln 2$

The natural logarithm of $2$ is:

- $\ln 2 = 0 \cdotp 69314 \, 71805 \, 59945 \, 30941 \, 72321 \, 21458 \, 17656 \, 80755 \, 00134 \, 360 \ldots$

This sequence is A002162 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

### Natural Logarithm: $\ln 3$

The natural logarithm of $3$ is:

- $\ln 3 = 1.09861 \, 22886 \, 68109 \, 69139 \, 5245 \ldots$

This sequence is A002391 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

### Natural Logarithm: $\ln 10$

The natural logarithm of $10$ is approximately:

- $\ln 10 \approx 2 \cdotp 30258 \, 50929 \, 94045 \, 68401 \, 7991 \ldots$

This sequence is A002392 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

## Also known as

The **natural logarithm** is sometimes referred to as the **Napierian logarithm** for John Napier, although this was not actually the logarithm he was famous for inventing.

## Also see

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

## Historical Note

The natural logarithm was discovered by accident by John Napier in around $1590$, evolving from his invention of the Napierian logarithm as a tool for multiplication of numbers by addition.

He had no concept of the notion of the base of a logarithm and certainly did not use Euler's number $e$.

## Sources

- 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $2 \cdotp 718 \, 281 \, 828 \, 459 \, 045 \, 235 \, 360 \, 287 \, 471 \, 352 \, 662 \, 497 \, 757 \, 247 \, 093 \, 699 \ldots$ - 1992: Larry C. Andrews:
*Special Functions of Mathematics for Engineers*... (previous) ... (next): $\S 1.2.2$: Summary of convergence tests (footnote) - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $2 \cdotp 71828 \, 18284 \, 59045 \, 23536 \, 02874 \, 71352 \, 66249 \, 77572 \, 47093 \, 69995 \ldots$ - 2008: Ian Stewart:
*Taming the Infinite*... (previous) ... (next): Chapter $5$: Eternal Triangles