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 (Figure 1).
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: