# Fundamental Logarithmic identity

## Formula

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

### Proof

$b$ is a variable and it is multiplied by itself $x$ times to represent their product $m$ in exponential notation.

$b^{\displaystyle x} = m$

According to the definition of logarithm, the same equation can also be written in logarithm form.

$\log_{b} m = x$

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

The value of $x$ is logarithm of $m$ to base $b$. Substitute the value of $x$ in $b^{\displaystyle x} = m$ equation to obtain the fundamental logarithmic identity.

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

#### Example

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

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

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

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

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