Growth rate of the functions

Question

I am trying to order functions according to their growth rate but I am stuck here. y=5^{n}, y=(n,)^{n} , y=(n^{2})!

5^{n} is exponential I am assume this is bigger but I cannot proof that. When looking at their graphs it seems like (n^{2}). is bigger I cannot understand. Help pls.

Answer 1

Take the logarithms. This will tel you work with the much more manageable functions (keep in mind the Stirling approximation log(n!) ~= n * log(n) ):

n * log(5)
n * log(n!) ~= n^2 * log(n)
n^2 * log(n^2) = 2 n^2 * log(n)

Can you continue?