Finding expression for variable x from equations like xlogx + y = 0

Question

After spending hours of trying to find an expression for variable P1 from the equation in the link, I have not been successful. I would be very grateful if someone could show the steps in rearranging the equation to find expression for P1. In general, how does one solve for say variable x in equations like xlog(x) + y = 0 ?

Thanks in advance!

Problem

Answer 1

To solve the equation $$x\log x+y=0$$ let $x=e^f $. Then $$ fe^f=-y \implies f=W(-y)\implies x=e^{W(-y)}=-\frac y {W (-y)}, $$ where $W(y)$ is the Lambert $W$-function.