Growth race
Three runners. One adds the same amount every step (a × n). One is the square (n²). One multiplies by the same factor every step (bn). Give the linear runner a huge head start and the exponential a tiny factor. See who wins anyway.
What's going on
Adding grows in a straight line. Squaring bends upward, but each step adds only a bit more than the last. Multiplying is different in kind: each step adds a fixed fraction of what you already have, so the additions themselves grow. Eventually that beats any polynomial, no matter how small the factor b or how big the head start a. Watch for the trap at the start: at n = 1 the exponential is b, which beats 1² = 1, so it begins in the lead. Then, unless b is bigger than 2, the square catches it. The crossing that matters is the later one, after which the exponential is never caught again.
Switch on the log scale. The exponential becomes a straight line, because a log turns multiplication into addition. The square becomes a curve that flattens. Straight on a log chart is the signature of exponential growth, which is how scientists spot it in epidemics, bank balances and bacteria.
Try b = 1.1, which is only 10% growth per step, and a = 50. At N = 40 the exponential looks hopeless: about 45 against 1,600 for the square and 2,000 for the line. Push N to 100 and watch it pass the line at n = 89 and the square at n = 96. Then imagine N = 200.