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I know that the only positive integer solution to $$a^b=b^a$$ is $a=2$ and $b=4$, but what about rational solutions, what examples are there? I did some work and found that for example if

$$\left(\frac{c}{d}\right)^{\LARGE\frac{f}{g}}=\left(\small\frac{f}{\normalsize g}\right)^{\LARGE\frac{c}{d}},$$

then $c^{df}=f^{cg}$ and $d^{df}=g^{cg}$.

Cheese Cake
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blademan9999
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    Check this: https://math.stackexchange.com/q/4029626/42969, in particular the last paragraph in https://math.stackexchange.com/a/4029652/42969. – Martin R Jun 12 '23 at 07:30

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