What is the solution in $x$ to $$\frac{\ln x}{x} = k ?$$
I suspect this has something to do with the Lambert W function, since that's used in solutions of the form $x\ln(x) = k$, but the Wikipedia definition of $W$ is a little too abstract for me to apply this to $\frac{\ln x}{x} = k$.
Hint If we change variables to $y = \frac{1}{x}$ and rearrange, we produce $$y \log y = -k , $$ which has precisely the form that the original question remarks can be handled by the Lambert W function.
NB that many people (myself included) would not consider the Lambert W function to be closed.