Solve for all $x$:$$x^x=4$$
If we take the $\ln$ of both sides, we get: $$x\ln x=\ln 4$$ Since, $x=e^{\ln x}$, we get: $$(\ln x)e^{\ln x}=\ln 4 \implies \ln x=W(\ln 4)\implies x=e^{W(\ln 4)}$$ Where $W(x)$ is the Lambert $W$ function. Now, based on my understanding, we should get: $$x=e^{W(\ln 4)}e^{2\pi ik}=e^{W(\ln4)+2\pi ik}\qquad\forall k\in\mathbb N$$ However, Mathematica is telling me that one of the complete solutions to $x$ should be: $$x=e^{W(2\pi ik+\ln x)}\qquad\forall k\in\mathbb N$$ Checking Mathematica's answer numerically shows that it is a wrong answer, so what gives?