# Base of Logarithm

## Definition

A number which considered as a parameter to transform any number into its multiplying factors is called base of the logarithm.

Any number can be expressed as multiplying factors of another number. It is possible when one number is converted into multiplying factors on the basis of second number.

Hence, the number which considered as a parameter to transform a particular number into its multiplying factors is called base of a logarithm when the relation between them is expressed in mathematics by logarithm.

### Representation

The base of logarithm is written as subscript after $\log$ symbol to express that on the basis of this number, a particular number can be expressed as multiplying factors.

For example, if $b$ is the base, then it is expressed as follows.

$\log_{\displaystyle \, b}$

After this only, the number is written.

#### Example

$256$ is a number and understand following three cases to know what exactly the base is in logarithms.

1
##### Express it in terms of $2$

$256 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2$

$\implies 256 = 2^{\displaystyle 8}$

Express relation between three of them by logarithm.

$\log_{\displaystyle 2} 256 = 8$

On the basis of number $2$, the number $256$ is expressed as eight multiplying factors of $2$. Therefore, the number $2$ is called base of the logarithm of $256$.

2
##### Express it in terms of $4$

$256 = 4 \times 4 \times 4 \times 4$

$\implies 256 = 4^{\displaystyle 4}$

Express the relation between three of them in logarithm system.

$\log_{\displaystyle 4} 256 = 4$

On the basis of number $4$, the number $256$ is written as four multiplying factors of $4$. Hence, the number $4$ is called as base of the logarithm of $256$.

3
##### Express it in terms of $16$

$256 = 16 \times 16$

$\implies 256 = 16^{\displaystyle 2}$

Write relation between three of them in logarithms.

$\log_{\displaystyle 16} 256 = 2$

On the basis of $16$, the number $256$ is expressed as two multiplying factors of $16$. Therefore, the number $16$ is called base of the logarithm of $256$.

The three cases clear that the number which considered as a parameter to transform a particular number into its multiplying factors is the base of the logarithm.

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