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?

log(infinity) < 100 for all practical purposes.

I am really curious for real world examples where this doesn't hold.

To clarify:

• I understand O(f(N))
• I am curious about real world examples where the asymptotic behaviour matters more than the constants of the actual performance.
• If log(N) can be replaced by a constant it still can be replaced by a constant in O( N log N).

This question is for the sake of (a) entertainment and (b) to gather arguments to use if I run (again) into a controversy about the performance of a design.