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If $x! = N^{\log N}\;,$ How can I estimate $x$ in terms of $N$?

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Stirling's formula might be a good start. –  Harald Hanche-Olsen Feb 20 '12 at 18:46
    
You can try Stirling's approximation formula for $x!$ and take logarithms (which will reduce the right hand side to $(\log N)^2$). –  Arturo Magidin Feb 20 '12 at 18:50

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If we had an approximation for the inverse gamma function $\Gamma^{-1}$, we could apply it to both sides to get $$x \approx \Gamma^{-1} \left( N^{\log{N}} \right)$$

An approximation of $\Gamma^{-1}$ can be found here.

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