One or Two Zeros

Let $f(x) = a - \dfrac{1}{x} - \ln x$ with $a \in \mathbb{R}$.

1. If $a = 2$, find the number of zeros of $f(x)$ on $\left(1, e^2\right)$. 2. If $f(x)$ has exactly one zero, find the set of possible values of $a$. 3. If $f(x)$ has two zeros $x_1 < x_2$, prove that