Solving for $y$ in $y = x \ln(y)$

Question

I want to solve $y = x \ln(y)$ for $y$ in terms of $x$. Wolfram Alpha kindly produces this plot with the solution, $y = -x W(-\frac{1}{x})$, where $W$ is the Lambert function.

However, that only finds $y$ on the bottom "half" of the curve. How do I find $y$ for the top half?

Answer 1

Here' you will want to use the other branch of the Lambert function, $W_{-1}(x)$. Recall that for $-1/e \le x < 0$, two of the infinitely many branches of the Lambert function take on real values: the principal branch $W(x)=W_0(x)$ and $W_{-1}(x)$. For $x \geq e$, then, your original expression can take the Lambert function to be any of those two real branches. Which branch to use would depend on your application, though...