A decent approximation to n! is sqrt(2pi n) * (n/e)^n. A worse approximation is that n! is about (n/e)^n.

So, if b(n) is the base in b^n=n!, then b(n) is about (n/e). [I see this as the key observation of the article.]

Keeping in mind that (n/e)^n is a bit smaller than n!, a more accurate value for b(n) is a little higher than n/e.

So it makes sense that 25! ~= 10^25 because 25/e is a bit below 10, and the adjusted value for b(n) ought to be a little higher than 25/e; 10 will do in this case.

(And 270! ~= 100^270, etc.)