# Base of Logarithm

A quantity which used to express any other quantity as its multiplying factors is called the base of logarithm.

## Introduction

Every quantity can be expressed as multiplying factors of another quantity in mathematics. It can be done by expressing a quantity as multiplying factors on the basis of another quantity.

The relation between the two quantities is represented by a logarithmic symbol $(\log)$ but the quantity which used to transform another number is displayed as subscript of $\log$ symbol for representing that quantity as a base of the mathematical operation.

### Example

$256$ is a quantity and let’s try to write this quantity on the basis another quantities to understand the importance of the base of logarithms.

#### Base 2

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

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

The relationship between three of them is expressed in logarithms.

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

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

#### Base 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. Hence, the number $4$ is called as base of the logarithm of $256$.

#### Base 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 as base of the logarithm of $256$.

The three cases are the best examples for understanding the concept of base of the logarithms.

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