Why does binary search have logarithmic time complexity?

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A three-dimensional Lorenz attractor graph with swirling blue lines forming a butterfly-shaped pattern, representing chaotic motion in a mathematical weather model, with labeled x, y, and z axes. — complexity The weather model of meteorologist Edward Lorenz
The x-, y-, and z-axes correspond to components of the weather, while the graph indicates feasible weather patterns. The graph is a function of time, which means, in theory, that tracing the curve in one direction reveals past weather patterns, while tracing the curve in the other direction predicts future weather patterns. However, practically indistinguishable points may lead to completely different weather epochs, indicated by the two distinct “lobes.” This phenomenon, known as sensitivity to initial conditions, is what prevents precise weather forecasts more than a few days into the future.

Binary search is classified as having logarithmic time complexity, denoted as $O (\log n)$ , because of how it processes data. Rather than examining each element individually, a binary search on a sorted list repeatedly divides the search space in half until the target element is located.1

Because the algorithm discards half of the remaining elements with each step, the number of operations required to find the target grows according to the logarithm of the input size $n$ (specifically in base 2), rather than growing proportionally to $n$ as in linear time algorithms. This logarithmic growth rate makes binary search significantly more efficient than linear search as the number of elements increases.1