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Having: $P = N^{CN + 1}$;

How can I simplify this equation to $N = \cdots$?

I tried using logarithms but I'm stucked...

Any ideas?

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up vote 3 down vote accepted

The equation $P=N^N$ doesn't have an elementary solution, but taking logarithms you can find approximate solutions (or even solutions in the form of transseries). In your case, take logarithms to find $(cn+1)\log n = p$. Assuming $p$ and so $n$ are large, $n\log n \approx p/c$. Therefore $n \approx p/c$, and so $$n \approx \frac{p/c}{\log (p/c)}.$$

Edit: the following is wrong, as J.M. pointed out. Again ignoring the $1$ in the exponent, we have $P^{1/c} = N^N$ and so $N = W(P^{1/c})$, where $W$ is the Lambert function (see Wikipedia).

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Um... no. The Lambert function is the inverse of $x\exp(x)$ (if $x=y\exp(y)$, then $y=W(x)$). You need to work a bit harder to invoke Lambert in inverting $x^x$. – J. M. Dec 12 '10 at 2:09

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