There are three types of change base rules in logarithms. Here is the list of changing base properties of logarithms in mathematical form with proofs.

1.

Log of a number to a base can be expressed in product form one multiplicative factor is log of the number to another base and the second multiplying factor is log of another number to actual base.

$\large \log_{b} m = \log_{a} m \times \log_{b} a$

2.

Logarithm of a number to a base can be expressed as a division of the logarithm of the number and logarithm of the base but both of them are having another base commonly in this base changing law.

$\large \log_{b} m = \dfrac{\log_{a} m}{\log_{a} b}$

3.

Logarithm of a number to a base can be written by swapping the number by base and base by number in reciprocal form in this fundamental change of base formula.

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

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