Asymptotic analysis of statements
The function $\log(n^n) = n\log n$ grows much faster than $\log n$, so we can conclusively say that $\log(n^n) = \omega(\log n)$. It even grows much faster than $n$, so $\log(n^n) = \omega(n)$.

You say that whether $\log(n^n) = \Theta(\log n)$ (where perhaps you mean $\Theta(n)$ instead of $\Theta(\log n)$) is "inconclusive", since for small $n$ there is one behavior, and for large $n$ there is another. For asymptotic analysis we care only about the behavior for large $n$.