For questions related to the Lambert W or product log function, the inverse of $f(z)=ze^z$.
Equations involving exponentials multiplied to linear terms, such as $$x+2=e^x$$, cannot be solved using elementary functions in general. However, they may be solved in closed form using the Lambert W function, the inverse of $$f(z)=ze^z$$. In the given example: $$x+2=e^x$$ $$(-x-2)e^{-x-2}=-e^{-2}$$ $$-x-2=W(-e^{-2})$$ $$x=-W(-e^{-2})-2$$ There are two real branches of the Lambert W: $$W_0(z)\text{ for }z\in[-1/e,\infty)$$ $$W_{-1}(z)\text{ for }z\in[-1/e,0)$$
There are also branches $$W_n(z)$$, with $$n\in\mathbb{Z}$$, which are defined only on the complex plane, but which still respect the property of solving $$W_n(z)e^{W_n(z)}=z$$.
The regions of the complex plane where the inverse relationship occurs are bounded by the curve $$x=-y\cot(y)$$.
As discussed in the Corless et al. reference, the use of $$W$$ follows early Maple usage. Exact solutions to some mathematical models in the natural sciences, such as Michaelisâ€“Menten kinetics, use this function.