O(log N) == O(1) - Why not?

Question

Whenever I consider algorithms/data structures I tend to replace the log(N) parts by constants. Oh, I know log(N) diverges - but does it matter in real world applications?

Answer 1

The rules of determining the Big-O notation are simpler when you don't decide that O(log n) = O(1).

As krzysio said, you may accumulate O(log n)s and then they would make a very noticeable difference. Imagine you do a binary search: O(log n) comparisons, and then imagine that each comparison's complexity O(log n). If you neglect both you get O(1) instead of O(log²n). Similarly you may somehow arrive at O(log¹⁰n) and then you'll notice a big difference for not too large "n"s.