## Logarithmic function

Logarithm with base . The logarithm of  with base , denoted by , is defined as the unique value  satisfying . In other word we have

 if and only if (1)

Immediate properties of logarithms are as follows:

Common and natural logarithms. The common logarithm of  (denoted by ) and the natural logarithm of  (denoted by ) are defined respectively by

and

(where )

Logarithmic function. Let be a positive real value. A function  is called an logarithmic function with base  if

When , is an increasing function and as . When , is a decreasing function and as . In either case, the logarithmic function  has the vertical asymptote . The domain  and the range  of  are given by and , respectively.

 1. 2.

Properties of logarithmic functions. The logarithmic function  is a one-to-one function; thus, implies . Furthermore, (2) indicates that is the inverse function  of the exponential function  .

Natural logarithmic function. A function  is called the natural logarithmic function if

Department of Mathematics