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

My answer here Fast random weighted selection across all rows of a stochastic matrix is an example where an algorithm with complexity O(m) is faster than one with complexity O(log(m)), when m is not too big.