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, $W_0(z)$ for $z\in[-1/e,\infty)$ and $W_{-1}(z)$ for $z\in[-1/e,0)$, as well as infinitely many complex ones.

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 Michaelis–Menten kinetics, use this function.