I find both the questions and answers confusing, I think this is a case when a little more "formal" understanding would help before jumping into "intuition" and big picture explanations.
If the question is really what O(log n) means, then the OP should first try to understand the definitions of Big-Oh notation, asymptotic growth rate, etc. At this stage it is premature to talk about running times and algorithms, these are simply statements about functions. Maybe the OP is already past this stage, or at least has a rough understanding of these concepts.
If the question is where does O(log n) arise in practice, then one can explain binary search, binary trees, the representation of a number in a certain base, harmonic series, etc.