TITLE: How big is a particular n!?
QUESTION [10 upvotes]: Is there a way to estimate how big $n!$ is for a certain $n$?
For example, without using a calculator, what is the magnitude of $7!$ or $12!$ or $100!$?

REPLY [4 votes]: A different tack here, as you I suspect you are asking for a method that doesn't require any 'advanced' functions.
How large a positive integer number is, can be expressed well in terms of how many digits it takes to write the number. This is best represented as the integer part of the logarithm of the number, plus 1 (logarithm base 10, i.e. Log not Ln). For example LOG[55] = 1.74 (2 d.p.), it requires INT[1.74] + 1 = 2 digits to represent it. You can get a better feel for the size of a number by looking at the digits after the decimal point in the logarithm too.
Doing polynomial fits of LOG[n!] vs n gives these two useful approximate working regions.
n = 7 to 119 (cubic polynomial): LOG[n!] ~ -0.0000284659105*n^3 + 0.00943671908*n^2 + 0.962859106*n - 3.95423829
Maximum error factor in range of 2.93
n = 120 to 1000 (5th order polynomial): LOG[n!] ~ -4.68977059E-13*n^5 + 0.00000000164012606*n^4 - 0.000002402405*n^3 + 0.0021512217*n^2 + 1.6770136*n - 29.6620872
Maximum error factor in range of 1.94
You could do better with higher order polynomials, particularly in range 7 to 119, but these are good enough to tell you "how big" the factorials are.
Below 7 you should be able to do in your head :-)