There is an infinite number of rational solutions
$$a=\left(\frac{n+1}{n}\right)^n,\;\;b=\left(\frac{n+1}{n}\right)^{n+1},\;\;n\in\mathbb{Z},\;\;0\neq n\neq -1.$$


For a proof that these are *all* the rational solutions of $a^b=b^a$ with $a\neq b$, see <A HREF="https://www.maa.org/sites/default/files/Sved50816668.pdf">Marta Sved's article</A> (1990). As she describes, this question has a long history, it was first proven by Euler and has been generalized in various ways.