# How to solve $x\log(x) = 10^6$

I am trying to solve

$$x\log(x) = 10^6$$

but can't find an elegant solution. Any ideas ?

Let $y=\log(x)$. Then the equation is $$ye^y=10^6.$$ The Lambert W-function is defined such that this means $y=W(10^6)$, and therefore $x=e^{W(10^6)}$. (This is effectively just a notational trick; it doesn't make anything more explicit).
You won't find a "nice" answer, since this is a transcendental equation (no "algebraic" solution). There is a special function related to this called the Lambert W-function, defined by $\ z = W(z) \cdot e^{W(z)} \$ . The "exact" answer to your equation is $\ x = e^{W( [\ln 10] \cdot 10^6)} \ .$ (I'm assuming you're using the base-10 logarithm here; otherwise you can drop the $\ln 10$ factor.)