exponential

Meaning: A quantity is exponential when its rate of change is proportional to its current value, so the variable sits in the exponent. Equal time intervals multiply the quantity by the same factor, which is why a half-life is constant. Decay drops fast at first and then levels off toward zero; growth accelerates. First-order kinetics in general chemistry has the same form.

Equation: N = N₀e^(−λt), where N is the amount at time t (same unit as N₀), N₀ is the initial amount, λ is the decay constant (s⁻¹), t is time (s), and e ≈ 2.718. Half-life: t½ = 0.693/λ.

On the MCAT: Common settings are radioactive decay, a capacitor discharging in an RC circuit (time constant τ = RC), and radiation intensity falling with absorber thickness. Expect to count half-lives (after three, 1/8 remains) or to read a graph: a straight line on a semi-log plot indicates exponential change.

Common confusion: Exponential is not the same as a power law such as x², where the variable is the base. Also, equal intervals remove equal fractions, not equal amounts, so two half-lives do not remove everything; one quarter remains.