Logarithmic Growth vs Linear Growth (Scaling comparison)

Logarithmic growth is much slower than linear growth. If $n = 1000000$, then $\log_2(n) \approx 20$. Linear search takes a million steps, binary search takes $20$.

Doubling $n$ from $1000000$ to $2000000$ increases linear time by $1000000$, but only increases logarithmic time by $1$ (from $20$ to $21$).

This is why $O(\log n)$ algorithms scale so well to huge inputs. Even with billions of elements, binary search takes fewer than $40$ steps.