# The Logarithmic Function.

### The Logarithm.

If “x” is a number and,

$x=a^y$
Then , “y” is known as the logarithm of “x” to the base “a”.

For example:
:-The logarithm of 16 to the base 2 is 4 which can be shown as:
$16=2^4$

:-The logarithm of 8 to the base 2 is 3 which can be shown as:
$8=2^3$

### The logarithmic Function:

The function in which the relationbetween input(let x) and and output(let y)

is given by:

$x=a^y$  , where “a” is a constant,

is known as the logarithmic function.

when,

$x=a^y$ ,

“y” is known as the logarithmic function of “x”  to the base “a”. and denoted as:

$log_{a}x=y$

### Relation between Exponential function and Logarithmic function:

Exponential function and Logarithmic are inverse function of each other if they are on the same base.

For example:

:- When ,

$8=2^3$

“8″ is the  Exponential function of “3″ and “3″ is the logarithmic function of “8″ to the base “2″.

Note:

:- Logarithmic function to the base 10 is known as common logarithm.

:- Logarithmic function to the base “e” is known as natural function where the value of “e” is given by:

$e=\lim_{x \to \infty}{1 + \dfrac{1}{x}}$

and if the base of a logarithmic function is “e” then the base is usually omitted and written as:

$log_{e}x=y$ = $log_{}x=y$

Related posts:

1. Exponential Function. Exponential function is the function in which the output is...
2. Inverse Function. If a function is defined from set A to set...
3. Composite Function. A single function formed by the combination of two or...
4. Exponent(index) and laws of exponents(indices). When , a number is repeatedly multiplied by itself the...
5. Types of Functions. In mathematics according to the nature shown by a Function...