If the factorial of a number N is 10^(10^6.935217102760750), then what is number N? This has stumped me! Thanks for any help.
First, one must have an arbitrary precision calculator that can handle very large exponents. Lucky for you, I have a couple of them!!.
Second, we have to evaluate this monster of a number:10^6.935217102760750=8,614,242.68476, which is the log of N! to base 10.
Next, we have to use "Stirling's Formula" for nery large N, which is:
n! ≈ n^n.e^−n.√2πn
Next, we can only handle such a large by taking the Logs of both sides. Since we already have the Log of this number, then by several iterations we hone in on the number. Thanks to some code in Java, which I wrote for this purpose, it zeroed in on the number very quickly!!!!. I plugged the number into the above Sterling's Formula and taking the Log of the RHS, it agrees almost exactly with the given Log. Therefore N=1,500,250. So that the Log of 1,500,250! agrees with a given Log of 8,614,242.68476.
This is a college-level problem, which is difficult to solve unless you are familiar with Sterling's Formula.
8,614,242.68476=n.Log(n).-n.Log(e).Log(√2πn)
If you sub 1,500,250 for n in the RHS, you will see they balance.
