is this conjecture true or false? I want to know if this conjecture istrue or false
$$\Large e^{\frac{ \ln x}{x}} \notin \mathbb{Z} $$
for every $x \in \mathbb{R} \setminus \{1,-1,0\} $

Hint
Firstly, show that as a real function, ${\rm dom}(f) = \mathbb R^{>0}$
Secondly, show that $f(\mathbb R^{>0}) = (0, e^{\frac1e})$ and conclude by
$$f(\mathbb R^{>0}) \cap \mathbb Z = \{1\}$$
and $f^{-1}(1) = \{1\}$
