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 had a function that takes:

1 millisecond to complete if you have 2 elements.
2 milliseconds to complete if you have 4 elements.
3 milliseconds to complete if you have 8 elements.
4 milliseconds to complete if you have 16 elements.
...
n milliseconds to complete if you have 2^n elements.

Then it takes log₂(n) time. The Big O notation, loosely speaking, means that the relationship only needs to be true for large n, and that constant factors and smaller terms can be ignored.