Differential equation solution with Lambert $W$ function.

Question

Solving the differential equation: $y'x\log y =1$ we easly find : $$ y(\log y-1)=\log x +c $$

I search an explicit solution $y=f(x)$ and WolframAlpha gives: $$ y=\dfrac{\log x+c}{W\left( \dfrac{\log x +c}{e}\right)} $$

Where $W$ is the Lambert function. I know that this function is defined such that $W(ze^z)=z$, but I don't see how this can give the Wolfram result.

Answer 1

Hint: Let $y=e^t$, then divide both sides with e.