How is the time complexity of the binary search algorithm determined?

The time complexity of the binary search algorithm is determined by the number of steps required to find an element in a sorted array.

The algorithm divides the search space in half at each step, reducing the number of considered elements by half. If the initial array contains n elements, then after one step there are n/2, after two — n/4, and so on.

The number of steps k needed to narrow down the search space to a single element satisfies the condition:

 n / (2^k) = 1

From this:

 2^k = n
 k = log2(n)

Thus, the time complexity of binary search is O(log n).

This means that the execution time grows logarithmically with the increase in input size, making binary search very efficient for large sorted arrays.