Monotonicity and Bolzano's theorem

Question

I have the function: $$f(x)=\frac{e^x}{\ln{x}},x\in(1,+\infty)$$ and I want to find the monotonicity of $f$. So: $$f'(x)=\frac{e^x(x\ln{x}-1)}{x(\ln{x})^2}$$ The sign of $f'$ depends on the sign of $x\ln{x}-1$. By Bolzano's theorem I can prove that $x\ln{x}-1=0$ has at least one solution $x_o$ ($x_0>1)$ and with monotonicity I can prove that is the only one. But I can find $x_0$ as a number, so how can I find the sign of $f'$ and therefore the monotonicity of $f$?

Answer 1

No closed form is possible with elementary functions. Using the Lambert function (inverse of $z\longmapsto ze^z$), the solution is $x = e^{W(1)}\approx 1.76322$. See Wolfram Alpha.