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Function $n(s)$ is given implicitly as $n! = s$. How can I find an asymptotic for $n(s)$? I cannot understand, how can I make an explicit function from this to calculate an asymptotic, as there is no reverse operation for factorial (I mean, like root is reversal to pow, and so on).

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See here. – Mhenni Benghorbal Oct 6 '12 at 4:54

Hint: Use the Stirling Approximation to the factorial. To get information about the growth of $n$ in terms of $s$, you will need asymptotic information about the Lambert function.

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From Stirling's approximation, you know that (for $n\to\infty$) $$s(n) \sim \sqrt{2\pi n} \left( \frac{n}e \right)^n.$$ The task is to invert this asymptotic relation.

Note that the dominant part reads $s\sim n^n$, i.e., $\log s \sim n \log n$ and $\log\log s \sim \log n + \log\log n$. As the first term dominates, we obtain the first approximation $$n(s) \sim \log s.$$

A better approximation reads $$n \sim \frac{\log s}{\log\log s}$$ and so on.

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