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 $W(x)$?
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2These are called super-roots and do not have closed forms using the Lambert W function. – Simply Beautiful Art Jan 10 '19 at 03:19
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@SimplyBeautifulArt, thank you for comment! How do you came to my question? – user514787 Jan 10 '19 at 03:23
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I've discussed this a bit, finding some generalization of the Lambert-W. See http://go.helms-net.de/math/tetdocs/Wexzal_Superroot.pdf – Gottfried Helms Jan 17 '19 at 11:36
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See this super root post – Тyma Gaidash Jun 12 '23 at 12:35