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Relation between Logarithms and Exponents

The logarithmic and exponential systems both have mutual direct relationship mathematically. So, the knowledge on the exponentiation is required to start studying the logarithms because the logarithm is an inverse operation of exponentiation.


The number $9$ is a quantity and it can be expressed in exponential form by the exponentiation.

$9 \,=\, 3 \times 3$
$\implies 9 \,=\, 3^2$

In this case, $3$ is a quantity and $2$ is the number of its multiplying factors. The inverse operation of $9 \,=\, 3^2$ is written in logarithmic form.

$\log_{3}{(9)} \,=\, 2$

The logarithmic system represents that the number of multiplying factors is $2$ when the quantity $9$ is written as multiplying factors on the basis of number $3$.

The mutual inverse mathematical relationship between exponential and logarithmic systems is written in mathematics as follows.

$9 \,=\, 3^2$ $\,\, \Leftrightarrow \,\,$ $\log_{3}{(9)} \,=\, 2$

Algebraic form

$y$ is a quantity. It is written in terms of $b$ and the total number of multiplying factors is $x$. The relation between three of them is written in mathematical form as per exponentiation.

$y \,=\, b^{\, \displaystyle x}$

It is written in logarithmic form in the following way to find the total number of multiplying factors by expressing y as multiplying factors of b..

$\log_{b}{(y)} \,=\, x$

The relationship between logarithmic and exponential systems is written in algebraic form as follows.

$y \,=\, b^{\, \displaystyle x}$ $\,\, \Leftrightarrow \,\,$ $\log_{b}{(y)} \,=\, x$

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