Is $O(\log_2 n)$ close to $O(n)$ or $O(1)$?
As far as I understand your question, you have to take a look at the differences: $O(\log n - 1) = O(\log n)$ versus $O(n - \log n)$. Since

\begin{eqnarray}
n - \log n &amp; \stackrel!&gt; &amp; \log n\\
n &amp; &gt; &amp; 2 \log n\\
\end{eqnarray}

you have $O(\log n) \subset{} O(n - \log n)$. The difference to $O(n)$ grows faster than the difference to $O(1)$.

Thus, $O(\log n)$ is closer to $O(1)$ than $O(n)$.