Change of Base Rule of Logarithm in Reciprocal form

Formula

$\log_{b} m = \dfrac{1}{\log_{m} b}$

Proof

According to logarithms, the log of any to number to same base is equal to one. Hence, the logarithm of a literal number $m$ to same literal is also one.

$\log_{m} m = 1$

The logarithm of $m$ to base $m$ can be written by using the change base rule of logarithm in product form. Here, the literal $b$ is used to change the base of the logarithm.

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

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

Therefore, it is proved that the reciprocal of the logarithm of $b$ to base $m$ is equal to logarithm of $m$ to $b$ and vice-versa. It is also called as the reciprocal property of change of base rule of the logarithm.

Verification

For example, the logarithm of number $25$ to base $5$ is $2$.

$\log_{5} 25 = \log_{5} 5^2 = 2$

Calculate the reciprocal of the logarithm of $5$ to base $25$ is also equal to $2$.

$\dfrac{1}{\log_{25} 5} = \dfrac{1}{\log_{5^2} 5} = \dfrac{1}{\dfrac{1}{2}} = 2$

$\therefore \,\,\,\,\,\, \log_{5} 25 = \dfrac{1}{\log_{25} 5} = 2$

The logarithmic identity has proved that the value of logarithm of a number to base is equal to its reciprocal.


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