TITLE: Can one find a closed form solution to $\ln x=\frac{1}{x}$,
QUESTION [5 upvotes]: Is there a closed form solution of the equation $\ln x=\frac{1}{x}$? I couldn't find a proof myself and I don't know any theorems that says when a closed form solution exists.

REPLY [5 votes]: The real solution is:
$$\ln(x)=\frac{1}{x}\Longleftrightarrow$$
$$x\ln(x)=1\Longleftrightarrow$$
$$e^{x\ln(x)}=e^{1}\Longleftrightarrow$$
$$x^x=e\Longleftrightarrow$$
$$x=e^{\text{W}(1)}$$
With $\text{W}(z)$ is the product log function.