A *logarithmic function* is a function that can be written in the form

Here,

is the base of the logarithmic function, and is positive number not equal to . The domain of a function of this form is . The equation is true if and only if .A *logarithm* is an expression of the form , and it represents the number that would need to be taken to the power of in order to equal . For example, is equal to .

The *natural logarithmic function* is the logarithmic function with base , but it is usually written with a different notation:

The natural log has many real-world applications, especially in growth and decay of natural systems and half-lives of isotopes.