Finding largest elements
The second loop in your solution requires time $O(\log n)$ and can be safely ignored since its time complexity is dominated by the previous sorting step.
The call to Quicksort requires time $O(n \log n)$ in expectation and $O(n^2)$ in the worst case.
You can solve your problem in $O(n)$ worst-case time as follows:
create a max-heap with the elements in the array. Then, perform $\log n$ extract-max operations.
Creating the heap takes $O(n)$ time while each of the maximum extractions requires $O(\log n)$ time, for a total time of $O(n + \log^2 n) = O(n)$.