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So far I couldn't find any related post of the title which is

Is there a nonzero real number $r$ such that $\pi^r$ is rational?

Is this a know problem? Or a corollary of some theorem?

Bill Dubuque
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    A much more difficult, to my knowledge unsolved, problem is whether or not there exist any such algebraic $x$. – FShrike Aug 16 '22 at 13:06

1 Answers1

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Hint: $x\mapsto \pi^x$ is a continuous function.

Or, if you want to go another way:

Hint:

Pick any rational number $q > 0$, and try to solve the equation $$\pi^x = q$$ by first taking the logarithm of both sides. Note that you can use the fact that $\log(a^b)=b\cdot \log(a)$.

5xum
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  • If we're being pedantic, the hint only works with certain definitions of log. I'm a bit doubtful that OP uses one of those definitions – Rushabh Mehta Aug 16 '22 at 13:00