Q: Does there exist an algebraic number $x,$ s.t. $f(x)$ is also an algebraic number?

$f(x)=\exp\bigg(\frac{1}{\ln(x)}\bigg)$ for $x\ne0,1.$

I would like to prove that the set of points $(x,f(x))$ is the empty set, for algebraic $x.$

Are there any promising approaches to solve this?