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

The logarithm of a nonzero positive number to the same quantity is called logarithm of base rule.

Logarithm of any number (nonzero positive number) is equal to one when the same number is taken as a base of the logarithm. There is a reason for this property. When a quantity is split as multiplying factors on the basis of same quantity, the total number of multiplying factors is one.

Therefore, it is called as logarithm of base rule. It is also simply called as log of base rule.

$m$ is a quantity and it is written as multiplying factors of another quantity $b$. Assume, the total number of multiplying factors of $b$ is equal to $x$.

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

Write product form of the quantity $m$ in exponential notation.

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

The quantity ($m$) in exponential form can be expressed in logarithmic form as per the mathematical relation between exponents and logarithms.

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

Write the mathematical relation between exponential form and logarithmic form of the quantity.

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

Assume, $x \,=\, 1$

$m \,=\ b^{\displaystyle 1}$ $\,\, \Leftrightarrow \,\,$ $\log_{b}{m} \,=\, 1$

The meaning of $b$ is raised to the power of $1$, is one time variable $b$.

$\implies m \,=\ b$ $\,\, \Leftrightarrow \,\,$ $\log_{b}{m} \,=\, 1$

When the value of $m$ is equal to $b$, then the logarithm of $m$ to the base $b$ is equal to one. In this case, $m \,=\, b$. Therefore, the variable $m$ can be replaced by $b$.

$\implies m \,=\ b$ $\,\, \Leftrightarrow \,\,$ $\log_{b}{b} \,=\, 1$

Therefore, it is proved that log of a quantity is equal to one when the same quantity is considered as a base of the logarithm.

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

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