Proving Upper Bound for Two Variable Function?

Question

The question is:

Prove (logn)^k = O(n) for every k>=1. 

I have never encounter a problem for proving an upper bound for two variables, so I am perplexed as to how I would approach solving this.

Answer 1

Misread the question, because it is (logn)^k and not log(n^k), which is what the following assumes.

By definition of Big O:

(logn)^k ≤ C*n, for N>No

Using a logarithm identity,

k*logn ≤ C*n, for N>No.

Thus, C = k and No = 0.

k*logn ≤ k*n for N>0

It is obvious that for any k≥1, that the above would hold true.

q.e.d.