Solving $x\log x=1$ without numerical methods

The exact solution requires the use of the Lambert W function. $$x\log x=1$$ $$(\log x)e^{\log x}=1$$ Now use the relation of the Lambert W function, $W(z)e^{W(z)}=z$: $$\log x=W(1)$$ $$x=e^{W(1)}=\frac1{W(1)}$$ $W(1)$ is sometimes called the omega constant, denoted as $\Omega$. This happens to be equivalent to the $\alpha$ you found with Newton's method.