What does O(log n) mean exactly?

Question

I am learning about Big O Notation running times and amortized times. I understand the notion of O(n) linear time, meaning that the size of the input affects the g

Answer 1

If you plot a logarithmic function on a graphical calculator or something similar, you'll see that it rises really slowly -- even more slowly than a linear function.

This is why algorithms with a logarithmic time complexity are highly sought after: even for really big n (let's say n = 10^8, for example), they perform more than acceptably.