# Natural Logarithm of a Number

## Natural Logarithm of a Number

Exponent to which the number e must be raised to obtain a given number.

The letter e represents the irrational number whose approximate value is 2.718 28.

The mathematician Euler defined this irrational number as being the limit of the following mathematical sequence.
$$e=1+\frac{1}{1}+\frac{1}{1\times2}+\frac{1}{1\times2\times3}+\frac{1}{1\times2\times3\times4}+\cdots =\sum_{n=0}^{\infty }\frac{1}{n!}$$

### Notation

If $$a = en$$, then $$n$$ is the logarithm of $$a$$ to the base e.
This relationship is written as n = loge(a) which is read as “n is equal to the logarithm of a to the base e”.
The logarithm to the base e of x is also written as ln x for “natural logarithm of x“.

### Examples

If loge(5.590) ≈ 1.721, then 5.590 ≈ e1.721.
If loge(12.566) ≈ 2.531, then 12.566 ≈ e2.531.

### HISTORICAL NOTE

This Napierian logarithm was named after Scottish mathematician John Napier, who created the first logarithmic tables in mathematics.
Leonhard Euler was the first to use the symbol e as the base of a system of natural logarithms in a letter written in 1731. In 1737, he proved that the number e is an irrational number. Then, in 1873, French mathematician Charles Hermite (1822-1901) proved that the number e is a transcendental number.