Exponential Growth & Decay Calculator

Exponential Models

Growth & Decay Formula

r > 0 for growth, r < 0 for decay

Half-Life Formula

T = half-life period

Newton's Law of Cooling

Common Applications

Half-Life Examples

• Carbon-14: 5,730 years
• Uranium-238: 4.5 billion years
• Iodine-131: 8 days
• Caffeine: ~5 hours

What Is Exponential Growth?

Exponential growth happens when a quantity changes by a constant percentage each period, so the change itself keeps growing. A population rising 5% per year does not add the same number of people each year — it adds 5% of an ever-larger total. This compounding effect is what makes exponential curves start slowly and then climb steeply.

Discrete vs. Continuous Growth

Growth can be applied once per period (discrete) or at every instant (continuous). The calculator above uses the continuous model:

The two are linked: a discrete rate r per period equals a continuous rate k = ln(1 + r). For small rates they are nearly identical — 5% discrete becomes k = ln(1.05) ≈ 0.04879 continuous.

Worked growth example: A bacteria culture starts at 500 cells and grows continuously at 30% per hour. After 5 hours: P = 500 · e^{0.3 × 5} = 500 · e^1.5 ≈ 500 × 4.4817 = 2,240.84 cells.

Doubling Time and the Rule of 72

Doubling time is how long a growing quantity takes to double. For continuous growth at rate r:

At 5% continuous growth, that is 0.6931/0.05 ≈ 13.86 years. The famous Rule of 72 gives a quick mental estimate: divide 72 by the percentage rate — 72/5 = 14.4 years, close to the exact discrete value of 14.21 years. The rule works because ln(2) ≈ 0.693, and 72 has many convenient divisors.

Exponential Decay and Half-Life

With a negative rate, the same formula describes exponential decay: quantities shrink by a fixed percentage each period. Decay is usually expressed through the half-life T — the time for half the substance to disappear:

Worked decay example: A sample of 100 g has a half-life of 5 days. After 15 days (three half-lives): N = 100 · (1/2)³ = 100/8 = 12.5 g. After another 5 days only 6.25 g remains — decay never quite reaches zero.

Newton's Law of Cooling

A hot object cools toward the surrounding temperature at a rate proportional to the temperature difference:

Where T_a is the ambient temperature, T₀ the initial temperature, and k the cooling constant. Worked example: Coffee at 90°C in a 20°C room with k = 0.05 per minute. After 10 minutes: T = 20 + 70 · e^−0.5 ≈ 20 + 70(0.6065) = 62.5°C. The same formula models warming — a cold drink left in a warm room follows it in reverse.

Solving for the Rate or the Time

Logarithms unlock the other variables. From P = P₀e^rt:

Worked example: An investment doubles in 10 years. What continuous rate does that imply? r = ln(2)/10 ≈ 0.0693 = 6.93% per year.

Real-World Applications

Common Mistakes to Avoid

• Entering the rate as a percent in e^rt: The continuous formula needs the decimal — 5% is 0.05, not 5.
• Mixing discrete and continuous rates: 5% discrete ≠ 5% continuous; convert with k = ln(1 + r).
• Mismatched time units: A "per hour" rate needs time in hours, not days or years.
• Assuming exponential when it is linear: Fixed amounts per period are linear; fixed percentages are exponential.
• Forgetting decay never hits zero: Half-life models approach zero asymptotically — (1/2)ⁿ is never exactly 0.

Comparing Growth Rates

Small rate differences compound into large gaps. Discrete doubling times at several rates:

Worked Example: Radioactive Dating

Carbon-14 has a half-life of 5,730 years. A sample containing 60% of its original carbon-14 is how old? Rearrange N = N₀(1/2)^t/T: t = T · ln(N/N₀) / ln(1/2) = 5,730 × ln(0.6)/ln(0.5) ≈ 5,730 × 0.7370 ≈ 4,223 years. Check: (1/2)^4223/5730 = (1/2)^0.737 ≈ 0.60 ✓

Worked Example: Population Projection

A city of 2 million grows at 1.5% per year (discrete). In 20 years: P = 2 × (1.015)²⁰ ≈ 2 × 1.3469 = 2.69 million. The same projection continuously uses k = ln(1.015) ≈ 0.01489: 2e^{0.01489 × 20} ≈ 2.69 million — the models agree to two decimals at small rates.

Beyond Pure Exponentials: The S-Curve

Real populations and markets rarely grow exponentially forever — limited resources bend the curve into an S-shape (logistic growth): fast growth at first, then a slowdown toward a carrying capacity. Exponential models are accurate for the early phase, which is exactly when most projections are made.

How Long to Grow Tenfold?

Doubling time generalizes to any multiple. The time to grow by a factor of m at continuous rate r is t = ln(m)/r. At 5% continuous growth, growing tenfold takes ln(10)/0.05 ≈ 46.05 years — a useful figure for long-range technology and investment projections.

Present Value: Discounting Backwards

Exponential math also runs in reverse. To have $50,000 in 15 years at 6% continuous growth, how much must you invest today? Rearrange P(t) = P₀e^rt:

This "discounting" is how bonds are priced and how pension funds value future obligations — the present value of money falls exponentially the further out it sits.

Finding the Half-Life from Two Measurements

You do not need to know the half-life in advance. If a sample falls from 100 g to 70 g in 5 days, the decay constant is λ = −ln(0.7)/5 ≈ 0.071335 per day, and the half-life is T = ln(2)/λ ≈ 9.72 days. Check: (1/2)^5/9.72 ≈ 0.70 ✓. Two data points are enough to characterize any exponential decay.