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\} $

You should ask this only for $x>0$, as the expression is not well defined otherwise. You can rule out the case $x\in(0,1)$ easily, since it implies $\ln(x)/x<0$. Now find the maximum of $\ln(x)/x$ on $(1,\infty)$, and conclude.
