# Definition:Natural Logarithm

## Definition

### Positive Real Numbers

The **(natural) logarithm** of $x$ is the real-valued function defined on $\R_{>0}$ as:

- $\ds \forall x \in \R_{>0}: \ln x := \int_1^x \frac {\d t} t$

### 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:

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

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 and third are ambiguous (it doesn't tell you which base it is the logarithm of).

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

## Examples

### Natural Logarithm: $\ln 2$

**Mercator's constant** is the real number:

\(\ds \ln 2\) | \(=\) | \(\ds \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^\paren {n - 1} } n\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds 1 - \frac 1 2 + \frac 1 3 - \frac 1 4 + \dotsb\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds 0 \cdotp 69314 \, 71805 \, 59945 \, 30941 \, 72321 \, 21458 \, 17656 \, 80755 \, 00134 \, 360 \ldots \ldots\) |

### Natural Logarithm: $\ln 3$

The natural logarithm of $3$ is:

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

### Natural Logarithm: $\ln 10$

The natural logarithm of $10$ is approximately:

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

## 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.

Some sources call it the **hyperbolic logarithm**.

## 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 $e$.

## Linguistic Note

The word **logarithm** comes from the Ancient Greek **λόγος** (**lógos**), meaning **word** or **reason**, and **ἀριθμός** (**arithmós**), meaning **number**.

## 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*(2nd ed.) ... (previous) ... (next): $\S 1.2.2$: Summary of convergence tests (footnote) - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $2 \cdotp 71828 \, 18284 \, 59045 \, 23536 \, 02874 \, 71352 \, 66249 \, 77572 \, 47093 \, 69995 \ldots$ - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**natural logarithm** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**natural logarithm** - 2008: Ian Stewart:
*Taming the Infinite*... (previous) ... (next): Chapter $5$: Eternal Triangles: The number $e$ - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**natural logarithm**