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Is there a way to solve for an unknown in a factorial?

I was just wondering, what would be the opposite of factorial?

For example, If I had $n! = 120$. How can I then show algebraically that $n = 5$?

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    You could repeatedly divide your number by increasing integers; and if at any point you divide by $k$ and are left with $k+1$ then you know that the number you started with is $(k+1)!$.2012-10-02
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    Are you looking for a solution that works when the result is a non-integer? For example, suppose someone asks you for $n$ such that $n! = 200$. Do you want to say "There is no such $n$," or do you want to say "$n$ would have to be between 5 and 6", or do you want to say "$n\approx 5.297$"?2012-10-02
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    @MJD If possible, I would like to know both methods.2012-10-02
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    [Relevant](http://mathoverflow.net/questions/12828/inverse-gamma-function)2012-10-02

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[Added because of a question in a comment] The generalization of the factorial is the gamma-function: $n! = \Gamma(1+n) $ where we can also insert noninteger values for n: $y = \Gamma(z) $ such that we have a function over the complex numbers $z$ except the poles at the non-positive integers).[/added]

The gamma-function has two real fixpoints. If you write the power-series of the gamma around one of that fixpoints, then this power series has no constant term and can be reverted by series-reversion. From this you can then get the inverse of the gamma, and from this the inverse of the factorial. Unfortunately, the convergence-radii of that series are both small, so I cant say at the moment, how useful this process would actually be.

(I think I've seen a question concerning the inverse of the gamma here or on MO, and possibly even showed a couple of that coefficients: see here for a short discussion)

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    I don't know what the gamma function is.. Can you please offer an explanation to what this means?2012-10-02
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    @gekkostate: you could start at [Wikipedia](http://en.wikipedia.org/wiki/Gamma_function) or [Mathworld](http://mathworld.wolfram.com/GammaFunction.html)2012-10-02
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    @gekkostate: you can also numerically invert [Stirling's approximation](http://en.wikipedia.org/wiki/Sterling%27s_approximation) that $\ln n! \approx n \ln n -n +\frac 12\ln(2\pi n)$ to get intermediate values2012-10-02
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    You are probably referring to this thread: http://mathoverflow.net/questions/12828/inverse-gamma-function2012-10-02