Have I properly sorted these runtimes in order of growth?

Question

I am doing this small task which I have to arrange asymptotic runtime in ascending order. Here are the runtimes:

Here is the order I believe they should go in:

log10(n^4), n^3, 2^((log4n)), 2^(100n), e^pi^4096, n! + 12^1000

Is this correct? Or are there any errors?

Thanks!

Answer 1

I don't believe that the ordering you've given here is correct. Here are a few things to think about:

1. Notice that 2^{log₄ n} = 2^{(log₂ n / log₂ 4)} = 2^{(log₂ n) / 2}. Can you simplify this expression?
2. How fast does the function e^π4096 grow as a function of n?

Hope this helps!