Question 1106986: 1. x^x=46656
2. x!=3628800
If I plug in some values, I can solve, but the problem is that I do not figure out a way to list out the possess. So what's the possess?
Answer by math_helper(2461)   (Show Source):
You can put this solution on YOUR website! Are you by chance studying a chapter about estimation and/or iterative solutions? Both these problems lend themselves to iterative ("guessing") methods.
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1. 
Since the left hand side changes rapidly with changes in x, I chose to re-write it by taking ln() of both sides:
 approx= 
Now the left side doesn't change very rapidly so as we get close the guesses won't change much:
4 ln(4) approx= 5.545 (way too low)
5 ln(5) approx= 8.047
6 ln(6) approx= 10.7505568
Note that this is exactly the same estimate we are looking for, checking  shows it is equal to  so the answer here is  . Note that the guessing method would have worked even if x was a real number, not just an integer.
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2. There is no easy inverse function for the factorial (gamma function).
However, given x! = 3628800 you can start by dividing by 2, then 3, then 4, etc. When your division produces a value of 1 then that last division is the value of x.
Example: n! = 24
n/2 = 12
12/3 = 4
4/4 = 1 <<< 4! = 24
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Another approach is to iterate (guess, and adjust your next guess up or down accordingly):
8! = 40320 (too low, try a larger value of x)
12! = 479001600 (too high, next try a smaller value of x)
So we already know 8 < x < 12, there are only 3 values left to try.
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