This is the third post in the Law of Logarithm Series, the first is the the product law of logarithms and the second is the quotient law of logarithms. In this short post, we prove the power law or the power rule of logarithms. That is, .
Let . This is equivalent to in exponent form. Raising both sides by , we have
Getting the logarithm of both sides, we have
Substitute to the value of , we have