Solve $x\log x=y$

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I have the following equation, $x\log x=y$.

Is it possible to solve $x$ in terms of $y$.

I think it is not possible but I am not sure.

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1 Answer

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$x$ must be positive, for the $\log$ to exist, so you can write $x=e^u$ and the equation becomes:

$$ue^u=y$$

This equation is well known, and has no solution in elementary function, but in terms of the Lambert W function:

$$u=\mathrm{W}(y)$$

Hence

$$x=e^{\mathrm{W}(y)}$$

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