Are there any cases where you would prefer a higher big-O time complexity algorithm over the lower one?

Question

Are there are any cases where you would prefer O(log n) time complexity to O(1) time complexity? Or O(n) to O(log n)?

Answer 1

In a realtime situation where you need a firm upper bound you would select e.g. a heapsort as opposed to a Quicksort, because heapsort's average behaviour is also its worst-case behaviour.