The logarithm of a number is the exponent by which another fixed value, the base, has to be raised to produce that number.
The natural logarithm is the logarithm with base equal to *e*.

The function slowly grows to positive infinity as x increases and rapidly goes to negative infinity as x approaches 0 ("slowly" and "rapidly" as compared to any power law of x); the y-axis is an asymptote, as seen in .

Also known as Euler's number, *e* is an irrational number representing the limit of:

as n approaches infinity.
In other words, *e* is the sum of 1 plus 1/1 plus 1/(1*2) plus 1/(1*2*3), and so on.

The number *e* has many applications in calculus, number theory, differential equations, complex numbers, compound interest, and more.
It also is extremely useful as a base in logarithms; so useful that the logarithm with base *e* has its own name (natural logarithm) and symbol.
Here is the proper notation for the natural logarithm of x:

The natural logarithm is so named because unlike 10, which is given value by culture and has minimal intrinsic use, *e* is an extremely interesting number that often "naturally" appears, especially in calculus.

The inverse of the natural log appears, for example, upon differentiating a logarithm of any base:

Outside of calculus, the natural logarithm can be used to relate 1, *e*, *i*, and *π*, four of the most important numbers in mathematics: