The basic approximation I use is
n! ~ (n/e)^n so ln(n!) ~ n(ln(n)-1).
The basic inequality is
(n/e)^n < n! < (n/e)^{n+1}.
When considering convergence of series using the n-th root test,
(n!)^{1/n} ~ n/e so
((an)!)^{1/n} = (((an)!)^{1/(an)})^a ~(an/e)^a = a^an^a/e^a.
