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The following question is from my colleague. It seems to be emerged considering some elementary number theory problem. This is NOT as exercise or problem in published material although it may looks like. Consider the following equation of positive integers $a$, $b$, $x$, $y$, and a real number $r$. $$ \log_a x= \log_b y =r \quad (\ast)$$ Can we expect some rigidity of the condition like the following

Suppose that ($\ast$) holds for some positive integers $a$, $b$, $x$, $y$ and an irrational number $r$. Then we must have that $\log_a b$ is rational.

At first I guess I can solve this shortly. But after trying some quick attacks I realized that this may be harder than it looks.

seoneo
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    If you had some failed attempts, it is worth mentioning them, to prevent others from repeating these. – lisyarus May 13 '19 at 10:36
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    I believe that implication must be false, since it implies that the log to base b of x, which equals the log to base a of x divided by the log to base a of b, is irrational, as the quotient of an irrational number by a rational one. But many counterexamples exist, e.g. any case where b and x are equal. – Simon May 13 '19 at 10:36
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    @Simon Good observed, but can you show that such a counterexample can actually satisfy $"(*)"$ ? – Peter May 13 '19 at 10:49
  • No idea whether it helps, but $r$ must be even transcendental, if $"(*)"$ holds. Otherwise $x=a^r$ would be transcendental because of the Gelfond-Schneider-theorem. – Peter May 13 '19 at 11:17
  • Good point Peter ! – Simon May 13 '19 at 11:18
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    In fact, the problem seems to be hard ! – Peter May 13 '19 at 11:39
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    This is essentially the four exponentials conjecture (https://en.wikipedia.org/wiki/Four_exponentials_conjecture). It is open even for $a = 2$ and $b = 3$. – user670344 May 13 '19 at 19:12
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    See https://mathoverflow.net/questions/17560/if-2x-and-3x-are-integers-must-x-be-as-well, and https://math.stackexchange.com/questions/3121572/is-there-such-an-x-that-both-2-fracx3-and-3-fracx2-are-simul/3132733#3132733 – user670344 May 13 '19 at 19:13
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    Many thanks to user670344. – seoneo May 14 '19 at 09:55

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