Calculation of the various values of the function shows that each is in the lead for a period of time, but the factoral function eventually wins.

[Aside: Creation of these numbers using a spreadsheet is an interesting task in itself]

n	log(100n)	n^100	100^n	n!
1	2	1	100	1
2	2.301	1.267E+30	10000	2
99	3.995	3.660E+199	1E+198	9.332E+155
100	4	1E+200	1E+200	9.332E+157
150	4.176	4.065E+217	1E+300	5.713E+262
268	4.428	6.508E+242	1E+536	9.172E+535
269	4.429	9.445E+242	1E+538	2.467E+538

Usually we say that O(n!) > O(a^n) > O(n^a) > O(log(an)) for large values of n.

Numerical calculation become difficult if we replace 100 by 1 million but progress in calculating the switch over points may be made using logarithms (here to base 10):

Clearly, this formula may easily be extended to other powers of 10.