When is ${\log(a) \over \log(b)}$ an integer?

Question

I've encountered this quite a bit.

If I have ${\log(a)\over \log(b)} = c$ where $b$ is a known positive integer, what can be said about $a$ if $c$ needs to be an integer?

Answer 1

Then $\ln a = c\ln b$, and taking the exponential on both side you get that $a= b^c$. That is, $a$ is a power of $b$.