The number of elements that can be sorted in $\Theta(\log n)$ time using heap sort is

Let $k$ be the number of elements to be sorted.

The time complexity of heap sort to sort $k$ elements is $\Theta(k \log k)$.

We are given that the sorting time is $\Theta(\log n)$, so:

Let us test option (C), where $k = \Theta\left(\frac{\log n}{\log \log n}\right)$:

Thus, $k = \Theta\left(\frac{\log n}{\log \log n}\right)$ elements can be sorted in $\Theta(\log n)$ time using heap sort. Option (C) is correct.