Differential equation solution with Lambert $W$ function.

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.

Hint: Let $y=e^t$, then divide both sides with e.
• From your hint I find: $\log y -1=W\left( \dfrac{\log x +c}{e}\right)$ , but how I can find the result of WolfranAlpha? – Emilio Novati Apr 29 '15 at 21:45