The inequality $$ x > y + z\log x $$ appears often when analyzing two objects with some different asymptotic rates, and I find myself staring at this "Lambert W" style of inequality very often in computer science and statistics.
For $y,z>0$, is there a clean solution for $x$ in terms of $y,z$, where a solution is a function $f(y,z)$ given analytically such that $f$ captures the asymptotic properties of $x$, i.e., $f(y,z)=O(\inf\{x>0|x > y + z\log x\})$?