Solution
= Solution
The <logarithm> laws give
$$
n\log_2(5n)+\log_2(n^2)=n\log_2 n+n\log_2 5+2\log_2 n.
$$
For $n\geq2$, both $n\leq n\log_2 n$ and $\log_2 n\leq n\log_2 n$. The expression is therefore at most $(3+\log_2 5)n\log_2 n$. Consequently
$$
\boxed{\text{True: }n\log_2(5n)+\log_2(n^2)=O(n\log_2 n).}
$$
The <Big O notation> concerns eventual behaviour; the zero value of $n\log_2 n$ at $n=1$ is irrelevant.