# 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

**Natural logarithms** are sometimes called **Napierian logarithms** (for John Napier, who pioneered them) although this name is rare nowadays.

## Also see

- Equivalence of Definitions of Real Natural Logarithm
- Properties of Natural Logarithm
- Results about
**logarithms**can be found**here**.