Prove $(log {(n)})^k = O(n)$ by definition, not by limit

Question

How can I prove $(log {(n)})^k = O(n)$ (for any constant $k > 0$) by definition, not by the limit?

By definition, that mean. there exists C, N such that: $$(log {(n)})^k = Cn$$ with $n > N.$ I tried by induction but fail.

My professor want to prove it without using limit method.

Answer 1

Here's a series of steps/hints that'll take you there:

1. Write $n = e^x$. Then you want to find a constant $C$ and $N$ so that for all $x \geq \log(N)$ we have $$x^k \leq C e^x\,.$$

2. Show that for all $x > 0$, $\frac{x^k}{k!} < e^x$.

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Edit: Misread the question, edited accordingly.