The inverse of the equation $y=x^x$ is $\frac{\ln(x)}{W(\ln(x))}$. It is clear that the answer to $x^x=4$ is $2$, but the expression $\frac{\ln(4)}{W(\ln(4))}$ is not evidently equivalent with $2$. How can the expression be simplified to prove it is equal to 2?
To be more clear, $W(x)$ is the inverse of the function $y=xe^x$, so $x=W(y)$.
In order to simplify your expression you require the following simplification rule for the Lambert W function $$\ln x = \begin{cases} \text{W}_0 (x \ln x), \quad x \geqslant \dfrac{1}{e},\\[2ex] \text{W}_{-1} (x \ln x), \quad 0 < x \leqslant \dfrac{1}{e}. \end{cases}$$ Note here $\text{W}_0 (x)$ denotes the principal branch for the Lambert W function while $\text{W}_{-1} (x)$ corresponds to the secondary real branch.
Now, for $\text{W}_0 (\ln 4)$, on applying the above simplication rule we have $$\text{W}_0 (\ln (4)) = \text{W}_0 (2 \ln 2) = \ln 2.$$ Thus $$\frac{\ln (4)}{\text{W}_0 (\ln (4))} = \frac{2 \ln 2}{\ln 2} = 2,$$ as expected.