Given a binary min-heap, getting a sorted array of the $\log n$ smallest elements
Consider another min-priority queue $q$. The result will be in array $a$.


Push the top of the heap into $q$.
For $i$ from $1$ to $\log n$:


Set $a_i = \text{pop}(q)$.
Push the two sons of $a_i$ in the heap into $q$.



$q$ may be implemented with a min-heap of size  $O(\log n)$, so the overall complexity is $O(\log n \log \log n)$.