Irrational logarithm criterion (${\log }_{a}b\in \mathbb{Q}⟺a^p=b^q\text{ for some }p,q\ge 1$)

For integers $a,b>1$, rationality of ${\log }_{a}b$ is equivalent to equality of some positive integer powers $a^p=b^q$. Exponentiation proves one direction and taking logarithms proves the other. Thus multiplicatively independent integers give an irrational number as their logarithm ratio. For distinct prime bases and arguments, unique factorization supplies that independence.