To be slightly nitpicky, it is problems that belong to complexity classes and not algorithms. Binary search is an algorithm that solves searching in a sorted list.
Funnily, although we usually think of it as having complexity O(log n), that does not hold true for the Turing machine model, with which the complexity classes P and NP are defined.
Funnily, although we usually think of it as having complexity O(log n), that does not hold true for the Turing machine model, with which the complexity classes P and NP are defined.