I just read the question Why does $\ln(x) = \epsilon x$ have 2 solutions?, and thought I'd point out a related area of investigation. The equation $\log x=\epsilon x$ has 2 solutions for $\epsilon>0$, and the first one admits a simple asymptotic form, $x=1+\epsilon+3\epsilon^2/2+\cdots$ . The interesting one is the solution near infinity, whose asymptotic form, to leading order, is
$x=\frac{1}{\epsilon}\left(\frac1\alpha+\beta(1-\alpha+\alpha^2)+\frac12\alpha^2\beta^2\right)+O(\epsilon),$
where $\alpha^{-1}=\log\epsilon-i\pi$ and $\beta=\log\alpha$. I've simplified this equation as much as possible, whose origin is Mathematica, but how in the world would one go about proving this? Note that the exact solution is $x=-\frac1\epsilon W_{-1}(-\epsilon)$, so this question is related to the asymptotics of the Lambert W function.