How to find out the probability of ordered pairs of rational or irrational or transcendental number $(a,b)$ such that $1<a<50$, $1<b<50$, and $\log_b a$ is rational?
Uniformly distribution is assigned.
This can only happen if $a$ and $b$ are both powers of the same number, so if both are taken from $2,4,8,16,32$, or if both are taken from $3,9,27$, or both from $5,25$, or both from $6,36$, or both from $7,49$. Can you work out the probability of that?
EDIT: I made the mistaken assumption that $a,b$ were to be integers. If they are real numbers, chosen independently and uniformly from $(1,50)$, then surely the probability that $\log_ba$ is rational is zero, simply because the rationals have measure zero in (any interval of) the reals.