I don't know how to find the inverse function of the function $f(x) = \ln(x)/x$, and I'm not even sure it exists? Would appreciate any help.
EDIT: I've talked to my professor and he has told me that I should only consider the function on the domain $x > e$. With this constraint it should be possible to find an inverse function on the domain. But how?
Best Answer
Write it as $$ -\frac{\log(x)}{x} = -\log(x) e^{-\log(x)} = -y$$ to see the relation with Lambert's $W$ function: $$-\log(x) = W(-y)$$ $$x = e^{-W(-y)}$$ Note that the $W$ function is multi-valued (to be expected by the remarks made in comments and other answers). For the domain $x\geq e$ you'll have to take the lower branch $W_{-1}$ on $[-1/e,0)$: $$x = e^{-W_{-1}(-y)}$$