[robconery]

I believe we're talking past each other. Big O has nothing to do with "actual run time*. It doesn't care what about the number of inputs you have - just that you have them.

Mathematically, if n=1000 then log n is 10. 10 operations vs. 1000 is, theoretically and time complexity wise, faster.

Our disconnect is "actual" vs. "theoretical" and I want to stress again that Big O is purely theoretical. It's a technical adjective.