# Definition:Natural Logarithm/Positive Real

## Definition

### Definition 1

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

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

### Definition 2

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

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

- $\ln x := y \in \R: e^y = x$

where $e$ is Euler's number.

### Definition 3

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

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

- $\ds \ln x := \lim_{n \mathop \to \infty} n \paren {\sqrt [n] x - 1}$

## 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 name is rare nowadays.

It needs to be noted that the **Napierian logarithm proper** was in fact a different construction.

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

## Also see

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

## Linguistic Note

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

## Sources

- 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**logarithm (log)** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**logarithm (log)**