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

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

(where
)

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

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