Proof of the power rule for logarithms What is the proof for the power rule for logarithms? And are there different ways to prove it?


$$\log_a(M^p)=p\log_a(M)$$

Proof : Let $t=\log_a(M)$. Then, by definition, we have $a^t=M$. So, we have
$$M^p=(a^t)^p=a^{pt}.$$
Hence, we have
$$\log_a(M^p)=\log_a(a^{pt})=pt=p\log_a(M).$$
