$e^{-x}-x=0$ solution procedure

Question

I realized that this exponential equation has to be solved using Lambert $w$ function, I also know that the result is $x= w(1)$, but I don't know how to get there. Would you mind helping me with this?

$$e^{-x}-x=0 $$

Answer 1

$$ e^{-x}=x$$ $$\frac{1}{e^x}=x$$ $$xe^x=1$$, which by definition x=W(1).

Definition: $$xe^x=C \rightarrow x=W(c)$$ explanation and wiki

:)

Answer 2

Note that this is a transcendental equation which would fit in nicely into the form from wherein you can use the Lambert $W$ function.$$\begin{align}e^{-x}=x \\ 1=xe^x \end{align} \\ x= W(1)$$ Also notice that $W(1)$ has a special value namely the Omega constant, $\Omega\approx0.56714329\ldots$