# Fundamental Law of Logarithms

## Formula

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

### 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 $x$.

$m \,=\, \underbrace{b \times b \times b \times \ldots \times b}_x \, factors$

$\implies m \,=\, b^x$

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

The quantity $m$ is expressed in exponential form as $b^x$ 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 $m \,=\, b^x$

$\implies m \,=\, b^x$

$\implies b^x} \,=\,$

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.

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

##### Examples

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

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

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

$(3) \,\,\,\,\,\,$ $4^\log_{4}{70}} = 7$

$(4) \,\,\,\,\,\,$ $19^\log_{19}{120}} = 12$

$(5) \,\,\,\,\,\,$ $317^\log_{317}{1000}} = 100$

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