Can one find a closed form solution to $\ln x=\frac{1}{x}$,

Question

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.

Answer 1

The (real) solution is $1/W(1)$ where $W$ is the Lambert W function.

Answer 2

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.