Lower-bound returns the smallest index i such that arr[i] >= target. The half-open interval [lo, hi) and the arr[mid] < target test are the two pieces that distinguish it from a standard binary search: when the midpoint is too small, the target must live strictly to the right; otherwise the midpoint itself remains a candidate. The loop terminates with lo === hi, and that index is the answer. Time complexity is O(log n). This is the C++ std::lower_bound and Python bisect.bisect_left rewritten in JavaScript.

Upper-bound is identical to lower-bound except the comparison is <= instead of <. That single change shifts the result from 'first index that is >= target' to 'first index strictly > target'. For values not present in the array, both functions return the same insertion point. The two together let you locate every occurrence of a value (and its multiplicity) in O(log n). The C++ analogue is std::upper_bound, the Python analogue is bisect.bisect_right.

Once you have both primitives, counting how many times a value appears in a sorted array reduces to subtracting their results. This pattern generalises to range queries: 'how many values fall in [a, b]' is upperBound(b) - lowerBound(a). Two log-n scans replace the linear pass a naive solution would do, which matters for repeated queries on a static sorted array. Pre-sort once, then answer each query in O(log n) with these two primitives.