Comparing asymptotic growth of logarithmic functions by reasoning

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As an exercise, we're sorting functions according to their asymptotical growth. When comparing these two functions, I'm getting stuck:

$n^2/(\log_2 n)^3$ versus $n \log_2 n$.

Using limits I am eventually able to figure out that the first one grows faster, but I'm actually looking for a way to do this without using limits.

How can I do that?