# Base of Exponential form

## Definition

A number which considered as a parameter to split any number as its multiplicative factors for expressing it in exponential notation is called base of the exponential notation.

Any number can be expressed as multiplicative factors on the basis of another number. The number which considered as a parameter and its number multiplicative factors are expressed in exponential notation to represent the respective number symbolically.

Due to splitting a number as multiplicative factors on the basis of a particular number to express it in exponential form, the number which is taken as a parameter to split the number as its multiplicative factors is known as base of the exponential notation. It is also called as the base of the exponential form.

### Example

$125$ is a number and express it as multiplicative factors of another number $5$.

$125 = 5 \times 5 \times 5$

Express this expansion in exponential notation.

$125 = 5^{\, \displaystyle 3}$

On the basis of number $5$, the number $125$ is expressed as three multiplicative factors of $5$. Hence, the number $5$ is called the base of the exponential notation for the number $125$.

Observe the following two examples for better understanding.

$2401 = 7 \times 7 \times 7 \times 7$
$\implies 2401 = 7^{\displaystyle \, 4}$

The number $7$ is called the base of the exponential form.

$59049 = 9 \times 9 \times 9 \times 9 \times 9$
$\implies 59049 = 9^{\displaystyle \, 5}$

The number $9$ is called the base of the exponential notation.

#### Algebraic form

Assume, $m$ is a number and it is split into $n$ number of multiplicative factors on the basis of another number $b$.

$m = \underbrace{b \times b \times b \times … \times b}_{\displaystyle n factors}$

Express relation between three of them in exponential notation.

$m = b^{\displaystyle n}$

On the basis of number $b$, the number $m$ is split into $n$ number of multiplicative factors. Therefore, the number $b$ is called base of the exponential form.

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