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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.

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What sort of distribution are you assigning to $(a,b)$? Are you considering $(a,b)$ to have a uniform distribution on the rectangle $\{\{0,0\},\{50,50\}\}$? –  Carl Morris Sep 24 '12 at 2:05
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1 Answer

up vote 2 down vote accepted

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.

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