The Power Law of Logarithms

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, \log_{p} A^r = r \log_p A.

Theorem

\log_p A^r = r \log A

Proof

Let \log_p A = x. This is equivalent to A = p^x in exponent form. Raising both sides by r, we have

A^r = p^{xr}.

Getting the logarithm of both sides, we have

\log_{p} A^r = xr.

Substitute \log_p A to the value of x, we have

\log_{p} A^r = r \log_p A. \blacksquare

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