# Definition:Natural Logarithm/Positive Real/Definition 2

## Definition

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.

That is:

- $\ln x = \log_e x$

## 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.

## Also see

- Equivalence of Definitions of Real Natural Logarithm
- 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

- 1960: Walter Ledermann:
*Complex Numbers*... (previous) ... (next): $\S 4.6$. The Logarithm - 1968: Murray R. Spiegel:
*Mathematical Handbook of Formulas and Tables*... (previous) ... (next): $\S 7$: Natural Logarithms and Antilogarithms - 1997: Donald E. Knuth:
*The Art of Computer Programming: Volume 1: Fundamental Algorithms*(3rd ed.) ... (previous) ... (next): $\S 1.2.2$: Numbers, Powers, and Logarithms: $(15)$