# Logarithm of any non-zero positive number to same base

## Formula

Logarithm of any nonzero positive number to same base is one.

For example, if $b$ is a number, then the logarithm of $b$ to same number is $1$.

$\log_{\displaystyle b} b = 1$

It is called as law of same base of logarithm and this logarithmic rule can be derived in mathematics from its definition.

### Proof

$m$ is a number and it is expressed as $x$ multiplicative terms on the basis of another number $b$.

$m = b^{\displaystyle x}$

The number of multiplicative terms of $m$ on the basis of $b$ is $x$.

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

Assume, the value of $x$ is $1$.

$m = b^{\displaystyle 1}$

$\implies \log_{\displaystyle b} m = 1$

In this case, the value of $m$ is equal to the value of $b$ if number of multiplying terms is one.

$m = b$

Now, replace $m$ by $b$.

$\therefore \,\, \log_{\displaystyle b} b = 1$

Similarly, replace $b$ by $m$

$\therefore \,\, \log_{\displaystyle m} m = 1$

It expresses that the logarithm of a number to same base is always $1$ mathematically and this logarithmic identity can be used as a formula in logarithms.

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