i want to Order these runtime complexitys ascending. I tried to transform this and made a first try in ordering them which you can find behind the -->
- $log4711$ --> 1.
- $log^4(n)$ --> 2.
- $log(n^4)$ --> $4log(n)$ --> 3.
- $\sqrt[4]{n}$ --> $n^{\frac{1}{4}}$ --> 4.
- $n^{\frac{3}{2}}$ --> 5.
- $9^{log_3(n)}$ --> $n^2$ --> 6.
- $e^{\frac{1}{3}n}$ --> 7.
- $(n-1)!$ --> 8.
I am not sure if my solution is correct and how to transform $log^4(n)$
Best Regards
As $\log n$ growth infinitely, $\log^4n$ growth faster than $\log n$, and so lines $2$ and $3$ should be swapped.
If you know that $\log n$ growth slower than any power, than to compare $\log^4$ and $\sqrt[4]n$ you can say that $\log n$ growth slower than $\sqrt[16]n$, and then take $4$th power of both sides.