Solving the equation $\ln(x)=-x$

Question

I tried solving this equation for a long time but did not succeed. Any help is appreciated.

$$\ln x=-x$$

I am not sure the tag is correct, I am not familiar with English mathematical terms. Please correct me if I am wrong.

Answer 1

This is $xe^x=1$, which means the solution is to use Lambert's W-function..

In this case, it is $W(1)$. This is also called the Omega constant.

Answer 2

To add to Thomas' answer, here are the steps $$ \ln x = -x $$ $$ e^{\ln x} = e^{-x} $$ $$ x = \frac{1}{e^{x}} $$ $$ xe^x = 1 $$ $$ x = W(1) $$ Where, $W$ is the Lambert W function.