# Fundamental Law of Logarithms

## Formula

$b^{\displaystyle \log_{b}{m}} \,=\, m$

### Proof

$m$ is a quantity and it is written in exponential form on the basis of another quantity $b$. The total multiplying factors of $b$ used to obtain the quantity $m$ is $x$.

$m \,=\, \underbrace{b \times b \times b \times \ldots \times b}_{\displaystyle x \, factors}$

$\implies m \,=\, b^{\displaystyle x}$

#### Express Quantity in Exponential form to Logarithm form

The quantity $m$ is expressed in exponential form as $b^{\displaystyle x}$ and it can be written in logarithmic form on the basis of mathematical relation between exponents and logarithms.

$\log_{b}{m} \,=\, x$

$\implies x \,=\, \log_{b}{m}$

#### Replace total factors in Exponential form equation

Earlier, it is taken that $m \,=\, b^{\displaystyle x}$

$\implies m \,=\, b^{\displaystyle x}$

$\implies b^{\displaystyle x} \,=\, m$

But it is written in logarithmic form as $x \,=\, \log_{b}{m}$. Now, substitute it in the exponential form equation to get the property of fundamental identity in logarithms.

$\therefore \,\,\,\,\,\, b^{\displaystyle \log_{b}{m}} = m$

##### Examples

Observe the following examples to understand the fundamental rule of logarithms.

$(1) \,\,\,\,\,\,$ $2^{\displaystyle \log_{2}{13}} = 13$

$(2) \,\,\,\,\,\,$ $3^{\displaystyle \log_{3}{5}} = 5$

$(3) \,\,\,\,\,\,$ $4^{\displaystyle \log_{4}{70}} = 70$

$(4) \,\,\,\,\,\,$ $19^{\displaystyle \log_{19}{120}} = 120$

$(5) \,\,\,\,\,\,$ $317^{\displaystyle \log_{317}{1000}} = 1000$