Solve $\arctan(x)-\ln(x)=0$ $f:(0, \infty)\rightarrow \mathbb{R}; f(x)=\arctan(x)-\ln(x).$
An interval $I$ in which the equation $f(x)=0$ has a real solution is: a. $(0,1)$; b.$(1,e)$; c.$(e,e^{2})$; d.$(e^{2}, \infty)$;
I tried to solve $f(x)=0$ with Rolle's Theorem but the derrivative has no real solutions.
Also, I tried with graphic method but I get stuck with intersection point of the functions.

For $x>0$, the function is indeed growing. Then
$$f(0)>0,$$
$$f(1)>0,$$
$$f(e)>0,$$
$$f(e^2)<0,$$
$$f(\infty)<0$$ gives you the answer.
