The lambert-W function \(x\mapsto xe^{x}\) turns up surprisingly often. For some reason I always have had a hard time approximating it. Today I finally found the “inequalities” section of the wikipedia page for Lambert-W function, and will share it here.
Theorem. Let \(W_0\) denote the real principal inverse of \(x\mapsto xe^{x}\). Then for any \(x\ge e\), \[W_0(x) = \ln x - \ln \ln x + \Theta\left(\frac{\ln\ln x}{\ln x}\right).\]
One related really common thing is the following: inverting \[x\ln x = y.\] Basically the inverse is approximately \(\frac{x}{\ln x}\). By which I mean, if \(f(x)=x\ln x, g(x) = \frac{x}{\ln x}\) then \[f(g(x))= x (1-o(1)).\]
Another useful taylor series to keep in mind: for small \(x\in (-\frac{1}{2}, \frac{1}{2})\) we have \[\log(1-x) \approx -x-\frac{x^2}{2}.\]