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

O(log n) is a bit misleading, more precisely it's O(log₂ n), i.e. (logarithm with base 2).

The height of a balanced binary tree is O(log₂ n), since every node has two (note the "two" as in log₂ n) child nodes. So, a tree with n nodes has a height of log₂ n.

Another example is binary search, which has a running time of O(log₂ n) because at every step you divide the search space by 2.