Exponential Growth & Decay

Model Setup

A = 1000 · e^(0.05 · 10)

A₀ (initial)

k (rate)

t (time)

k > 0 for growth, k < 0 for decay. t must be positive.

Examples

t = ln(200/100) / 0.693

A₀ (initial)

A (target)

k (rate)

A must be positive. For half-life: A = A₀/2, k < 0.

Examples

Result

Enter values above and press Calculate to see the result, growth/decay classification, and half-life or doubling time.

Growth

Final Amount (A)

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Half-life / Doubling Time

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Growth vs Decay

A = A₀ · eᵏᵗ

k > 0 (growth): The curve rises exponentially. The larger k is, the faster the growth. Population increases, investments compound.

k < 0 (decay): The curve falls toward zero asymptotically — it never quite reaches 0. Radioactive material, medication concentration, cooling.

Half-life (decay): the time for A to halve — t₁/₂ = ln(2) / |k|

Doubling time (growth): the time for A to double — t_d = ln(2) / k

Both formulas come from setting A = 2·A₀ or A = A₀/2 in A = A₀eᵏᵗ and solving for t using natural log.

Real-World Applications

A = A₀ · eᵏᵗ