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

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 would rephrase this as 'height of a complete binary tree is log n'. Figuring the height of a complete binary tree would be O(log n), if you were traversing down step by step.

I cannot understand how to identify a function with a logarithmic time.

Logarithm is essentially the inverse of exponentiation. So, if each 'step' of your function is eliminating a factor of elements from the original item set, that is a logarithmic time algorithm.

For the tree example, you can easily see that stepping down a level of nodes cuts down an exponential number of elements as you continue traversing. The popular example of looking through a name-sorted phone book is essentially equivalent to traversing down a binary search tree (middle page is the root element, and you can deduce at each step whether to go left or right).