How to prove that $2^x,3^x,5^x\in\mathbb N$ implies $x\in\mathbb N$?

Question

Let $x\in\mathbb R$ and suppose that $2^x,3^x$ and $5^x$ are all integers. Does it imply that $x$ is also necessarily an integer?

I read somewhere that the answer is "Yes" and a proof is known, but I haven't seen a proof and was not able to prove it myself. Could you please help me to find a proof?

Answer 1

See MO

It follows from the Six Exponentials Theorem. Apparently.