Function characterized by a relation in the form

*f*(*x*) = log\( _{a}\)(*x*) where*a*is a strictly positive real number that is different from 1.### Properties

Exponential functions of base *a* defined by *f*(*x*) = \({a}^{x}\) and logarithmic functions of base a defined by *f*(*x*) = log\( _{a}\)(*x*) are the inverse of one another.

If *a* > 1, the function defined by *f*(*x*) = \({a}^{x}\) is strictly increasing in the set of strictly positive real numbers and if 0 < *a* < 1, it is strictly decreasing in the set of strictly positive real numbers.

### Example

The function *f* defined in the set of real numbers by the relation *f*(*x*) = log\( _{2}\)(*x*) is a logarithmic function with base 2.

A function f defined in the set of real numbers by the relation *f*(*x*) = log\( _{3}\)(*x*) is a logarithmic function with base 3.