What does O(log n) mean exactly?

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 growth of the algorithm proportionally…and the same goes for, for example, quadratic time O(n²) etc..even algorithms, such as permutation generators, with O(n!) times, that grow by factorials.

For example, the following function is O(n) because the algorithm grows in proportion to its input n:

f(int n) {
  int i;
  for (i = 0; i < n; ++i)
    printf("%d", i);
}

Similarly, if there was a nested loop, the time would be O(n²).

But what exactly is O(log n)? For example, what does it mean to say that the height of a complete binary tree is O(log n)?

I do know (maybe not in great detail) what Logarithm is, in the sense that: log₁₀ 100 = 2, but I cannot understand how to identify a function with a logarithmic time.