$b^{\displaystyle \log_{b}{m}} \,=\, m$
$m$ 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 $x$.
$m \,=\, \underbrace{b \times b \times b \times \ldots \times b}_{\displaystyle x \, factors}$
$\implies m \,=\, b^{\displaystyle x}$
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.
$\log_{b}{m} \,=\, x$
$\implies x \,=\, \log_{b}{m}$
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.
$\therefore \,\,\,\,\,\, b^{\displaystyle \log_{b}{m}} = m$
Observe the following examples to understand the fundamental rule of logarithms.
$(1) \,\,\,\,\,\,$ $2^{\displaystyle \log_{2}{13}} = 13$
$(2) \,\,\,\,\,\,$ $3^{\displaystyle \log_{3}{5}} = 5$
$(3) \,\,\,\,\,\,$ $4^{\displaystyle \log_{4}{70}} = 70$
$(4) \,\,\,\,\,\,$ $19^{\displaystyle \log_{19}{120}} = 120$
$(5) \,\,\,\,\,\,$ $317^{\displaystyle \log_{317}{1000}} = 1000$
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