# Bounding Summations

I have an exam coming up that I'm studying for and I'm somewhat stumped by a couple problems. I need to derive an asymptotically tight bound for the following summation:

Assume $r$ and $s$ are constants such that $r,s\geq 0$.

$$f(x)=\sum_{k=1}^nk^r\log^s(k)$$

So far I have come up with

$$0\leq f(x)\leq(\sum_{k=1}^nn^r)(\sum_{k=1}^n\log^s(n))$$

$$f(x)\leq(n^{r+1})(n\log^s(n))$$

$$f(x)\leq(n^{r+2})(\log^s(n))$$

However, I'm getting told these are not asymptotically tight bounds and I need to correct this answer. Can anyone assist?

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