*e*must be raised to obtain a given number.

*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* = log_{e}(*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 log_{e}(5.590) ≈ 1.721, then 5.590 ≈ e^{1.721}.

If log_{e}(12.566) ≈ 2.531, then 12.566 ≈ e^{2.531}.

### 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.