= Irrational logarithm criterion
{title2=$\log_a b\in\mathbb Q\iff a^p=b^q\text{ for some }p,q\geq1$}
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.
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