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

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