# Solutions to $x \exp(x) = a x + b$

Are there any closed form solutions to $$x \exp(x) = a x + b$$ for real-valued $x$, $a$, and $b$ (we wish to solve for $x$, and $a$ and $b$ are simple constants)? We can assume that everything is positive.

I know that the solution to $$x \exp(x) = b$$ can be expressed via the Lambert $W$ function, and $$x \exp(x) = a x$$ is a simple log, but I can't get anything out of there.

• The solution to the last equation is simple log in addition to $x=0$. Don't forget that. – Arthur Aug 10 '16 at 22:09
• I suspect no closed form using $W$, but cannot prove. Here is very tangentially related. – 6005 Aug 11 '16 at 5:48
• If $b\approx0$, or close enough to $0$ where $a$ "dominates" the value of the LHS, then the solution is almost $x\approx\ln(a)$. If $a\approx0$, then the same argument gives $x\approx W(b)$. Not to much you can do if neither are the case. – Simply Beautiful Art Aug 17 '16 at 22:25