Difference of roots

Let function $u(x)=\dfrac{a}{x}-\ln x$ , where $x \in (0,\infty)$ , $a$ is a parameter and $a \in (e+1,2e)$ .

Is it always true that $f(x)=|u(x)|-\dfrac{\ln x}{x}$ will always two roots $x_1,x_2$ and $|x_1-x_2|<e$ ?

Bonus: Can the result be extended to $a \in (0,\infty)$ ?