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

Logarithmic running time (O(log n)) essentially means that the running time grows in proportion to the logarithm of the input size - as an example, if 10 items takes at most some amount of time x, and 100 items takes at most, say, 2x, and 10,000 items takes at most 4x, then it's looking like an O(log n) time complexity.