# Solve $x=d^{d}\log(d)$ using Lambert $W(x)$

We have $$a=b^b$$, so $$\log(a)=b\log(b)$$ $$x=\frac{x}{W(x)}\log\left(\frac{x}{W(x)}\right)$$ $$b=\frac{\log(a)}{W(\log(a))}$$ Next we have $$c=d^{(d^{d})}$$, so $$\log(c)=d^{d}\log(d)$$ In general $$^{k}m=n$$, $$^{k-1}m\log(m)=\log(n)$$ Is there a way to solve it as $$m=f_{k}(n)$$ using Lambert $$W(x)$$?