📈 Exponential Growth Calculator
Solve both standard exponential growth and decay (A = A₀(1+r)ᵗ) and continuous exponential growth and decay (A = A₀eʳᵗ) for any unknown variable. Generates a values table, interactive graph, doubling time, half-life, and full step-by-step working for every calculation.
🧮 Standard Exponential Solver — A = A₀(1+r)ᵗ
Labels your output
📋 Values Table Generator
| Time | Amount | Change from Prev. | % Change | Cumulative % Change |
|---|
📉 Interactive Exponential Graph
Exponential curve
Linear comparison
Doubling time / half-life
⏱️ Doubling Time & Half-Life Calculator
⚡ Rule of 72
Doubling Time ≈ 72 ÷ (Rate %)
Quick mental estimate — for 6% growth: 72 ÷ 6 = 12 periods (exact: 11.9). For 9%: 72 ÷ 9 = 8 periods (exact: 8.04).
| Growth Rate | Doubling Time (Exact) | Rule of 72 |
|---|---|---|
| 1% | 69.66 periods | 72 periods |
| 2% | 35.00 periods | 36 periods |
| 3% | 23.45 periods | 24 periods |
| 5% | 14.21 periods | 14.4 periods |
| 6% | 11.90 periods | 12 periods |
| 7% | 10.24 periods | 10.3 periods |
| 10% | 7.27 periods | 7.2 periods |
| 12% | 6.12 periods | 6 periods |
⚛️ Continuous Exponential Solver — A = A₀eʳᵗ
⚖️ Continuous vs Discrete Compounding Comparison
| Method | Formula | Result | vs. Continuous |
|---|
📐 Mathematical limit: As compounding frequency n → ∞, A₀(1+r/n)^(nt) → A₀eʳᵗ. More frequent compounding always yields slightly more, converging to the continuous limit.
🌍 Exponential Population Growth — P(t) = P₀eʳᵗ
Population at t
—
people
Doubling Time
—
years
| Year | Population | Increase from Start |
|---|
📚 Malthusian Growth Model: This exponential formula P(t) = P₀eʳᵗ is also called the Malthusian model (proposed by Thomas Malthus, 1798). It assumes unlimited resources, making it accurate for short-term projections but not long-term ones where carrying capacity limits growth.
📈 Continuous Growth / Decay Graph
Standard vs Continuous Exponential Growth — Which Formula to Use?
Use the standard formula when growth happens in distinct steps (annual population census, yearly compound interest). Use the continuous formula for natural processes that grow or decay at every instant (radioactive decay, bacterial growth, continuous compounding).
| Property | Standard (Discrete) | Continuous |
|---|---|---|
| Formula | A = A₀(1+r)ᵗ | A = A₀eʳᵗ |
| Compounding | Once per time period | Infinitely often |
| Base | (1+r) | e = 2.71828… |
| Rate meaning | % change per period | Instantaneous rate |
| Best for | Finance, population census data | Radioactive decay, continuous processes |
| When r is small | Very similar results | Very similar results |
Exponential Growth Formula — A = A₀(1+r)ᵗ Explained
The standard exponential growth formula A = A₀(1+r)ᵗ describes growth that multiplies by the same factor each time period.
A = A₀(1+r)ᵗ
A₀ = A / (1+r)ᵗ
r = (A/A₀)^(1/t) − 1
t = ln(A/A₀) / ln(1+r)
A — final amount (what you're solving for in most cases)
A₀ — initial amount (starting value, also written P₀)
r — growth rate as decimal (5% → r = 0.05)
t — number of time periods (years, months, days)
(1+r) — the growth factor (if r=0.05, factor = 1.05)
A₀ — initial amount (starting value, also written P₀)
r — growth rate as decimal (5% → r = 0.05)
t — number of time periods (years, months, days)
(1+r) — the growth factor (if r=0.05, factor = 1.05)
Intuition: Each period, multiply by (1+r). After t periods you've multiplied t times → (1+r)ᵗ
Period 0: A₀
Period 1: A₀ × (1+r)
Period 2: A₀ × (1+r)²
Period t: A₀ × (1+r)ᵗ
Period 0: A₀
Period 1: A₀ × (1+r)
Period 2: A₀ × (1+r)²
Period t: A₀ × (1+r)ᵗ
| Example | Formula Used | Calculation | Answer |
|---|---|---|---|
| $5,000 at 7%, 20 years | A = A₀(1+r)ᵗ | 5,000 × (1.07)²⁰ = 5,000 × 3.8697 | $19,348 |
| 250,000 people, 3%, 15 yr | A = A₀(1+r)ᵗ | 250,000 × (1.03)¹⁵ = 250,000 × 1.5580 | 389,494 people |
| 100,000→140,000 in 8 yr | r = (A/A₀)^(1/t)−1 | (1.4)^0.125 − 1 = 1.0432 − 1 | 4.32% per year |
| $1,000→$5,000 at 8% | t = ln(A/A₀)/ln(1+r) | ln(5)/ln(1.08) = 1.6094/0.07696 | 20.9 years |
Exponential Decay Formula — A = A₀(1−r)ᵗ Explained
The decay version uses A = A₀(1−r)ᵗ for discrete decay and A = A₀ × e^(−rt) for continuous decay. When r = 0.10, you lose 10% per period → decay factor = (1 − 0.10) = 0.90.
⚠️ Key insight: A 10% annual decay does NOT mean the quantity reaches zero after 10 years. Because each period you lose 10% of whatever remains (not 10% of the original), the curve approaches zero asymptotically — it never quite reaches it.
| Scenario | Values | Calculation | Result |
|---|---|---|---|
| Car depreciation, 15%/yr, 5 yr | A₀=$25,000, r=0.15, t=5 | 25,000 × (0.85)⁵ = 25,000 × 0.4437 | $11,092 (44.4% remains) |
| Radioactive decay, 5%/yr, 10 yr | A₀=200g, r=0.05, t=10 | 200 × (0.95)¹⁰ = 200 × 0.5987 | 119.7 grams |
| Same, 50 years | A₀=200g, r=0.05, t=50 | 200 × (0.95)⁵⁰ = 200 × 0.0769 | 15.4 grams |
Continuous Exponential Growth Formula — A = A₀eʳᵗ Explained
The continuous growth formula A = A₀eʳᵗ uses Euler's number e = 2.71828182845… Euler's number appears naturally whenever a quantity grows proportional to its current size.
A = A₀eʳᵗ
e = 2.71828…
r = ln(A/A₀) / t
🔢 Why e? The limit definition: as n → ∞, (1 + 1/n)ⁿ → e. If something grows 100% continuously: (1 + 1/∞)^∞ × start = e × start. Daily-compounding bank accounts approximate this continuous limit.
| Example | Formula | Calculation | Result |
|---|---|---|---|
| Bacteria, r=200%/hr, 2 hrs | A = A₀eʳᵗ | 100 × e^(2.0×2) = 100 × e⁴ | 5,460 bacteria |
| Carbon-14 decay, 5,730 yr | A = A₀e^(−rt) | 50 × e^(−0.0001209×5730) = 50 × 0.5001 | ≈ 25 g (exact half ✓) |
| Find r: 1,000→8,000 in 6 hrs | r = ln(A/A₀)/t | ln(8)/6 = 2.0794/6 | 0.3466/hr = 34.66%/hr |
⚗️ First Order Kinetics Connection: A = A₀eʳᵗ is identical to the first order kinetics equation [A] = [A]₀ × e^(−kt). Whether you're calculating radioactive decay, drug metabolism, or chemical reaction rates, it's the same exponential mathematics.
See our Reaction Order Calculator →
Doubling Time and Half-Life — Rule of 72 and Exact Formulas
Doubling time and half-life are mirror concepts — the same mathematics applied in opposite directions. The doubling time tells how long until a growing quantity doubles; the half-life tells how long until a decaying quantity halves.
t_d = ln(2)/ln(1+r)
t_d = 0.693/r (continuous)
t_h = ln(2)/|r| (continuous)
t_d ≈ 72/r%
⚡ The Rule of 72 — Quick Mental Estimate
Doubling Time ≈ 72 ÷ Rate %
6% growth → 72/6 = 12 periods (exact: 11.90) | 9% growth → 72/9 = 8 periods (exact: 8.04) | 12% growth → 72/12 = 6 periods (exact: 6.12)
⚗️ Chemistry connection: The half-life formula t½ = 0.693/k from first order chemical kinetics is identical to the continuous exponential decay formula. Whether you're studying radioactive decay or chemical reactions, the same mathematics applies.
Reaction Order Calculator →
Real-World Examples of Exponential Growth and Decay
📈 Growth Examples
👥 Population Growth
A city of 500,000 growing at 2% annually: after 25 years P = 500,000 × (1.02)²⁵ = 820,177 people. Exponential growth models cities, countries, and bacterial colonies where each individual contributes to reproduction.
A city of 500,000 growing at 2% annually: after 25 years P = 500,000 × (1.02)²⁵ = 820,177 people. Exponential growth models cities, countries, and bacterial colonies where each individual contributes to reproduction.
💰 Compound Interest
$10,000 at 7% for 30 years: A = 10,000 × (1.07)³⁰ = $76,123. Einstein reportedly called compound interest the "eighth wonder of the world" — its exponential nature means doubling time stays fixed regardless of balance size.
$10,000 at 7% for 30 years: A = 10,000 × (1.07)³⁰ = $76,123. Einstein reportedly called compound interest the "eighth wonder of the world" — its exponential nature means doubling time stays fixed regardless of balance size.
💻 Moore's Law
Transistor count doubles every ~2 years: classic exponential growth. From ~2,300 transistors in 1971 to 80+ billion in 2023, this is 52 years ÷ 2 = 26 doublings → factor of 2²⁶ = 67 million.
Transistor count doubles every ~2 years: classic exponential growth. From ~2,300 transistors in 1971 to 80+ billion in 2023, this is 52 years ÷ 2 = 26 doublings → factor of 2²⁶ = 67 million.
🦠 Bacterial Growth
E. coli doubles every 20 minutes. Starting from 1 cell: after 8 hours (24 doublings) = 2²⁴ ≈ 16.7 million cells. A = A₀eʳᵗ where r = ln(2)/0.333 hr = 2.08 per hour.
E. coli doubles every 20 minutes. Starting from 1 cell: after 8 hours (24 doublings) = 2²⁴ ≈ 16.7 million cells. A = A₀eʳᵗ where r = ln(2)/0.333 hr = 2.08 per hour.
📉 Decay Examples
☢️ Radioactive Decay
Carbon-14 has a half-life of 5,730 years. 100g of C-14 after 11,460 years: A = 100 × e^(−0.0001209×11460) = 100 × 0.25 = 25g. Used to date archaeological remains up to ~50,000 years old.
Carbon-14 has a half-life of 5,730 years. 100g of C-14 after 11,460 years: A = 100 × e^(−0.0001209×11460) = 100 × 0.25 = 25g. Used to date archaeological remains up to ~50,000 years old.
💊 Drug Metabolism
Ibuprofen has a half-life of ~2 hours. After taking 400mg: after 6 hours A = 400 × (0.5)³ = 50mg remains. Doctors use exponential decay to calculate safe dosing intervals for medications.
Ibuprofen has a half-life of ~2 hours. After taking 400mg: after 6 hours A = 400 × (0.5)³ = 50mg remains. Doctors use exponential decay to calculate safe dosing intervals for medications.
🚗 Asset Depreciation
A $30,000 car depreciating 20% annually: after 5 years A = 30,000 × (0.80)⁵ = $9,830 — only 32.8% of original value. Exponential depreciation means steeper losses in early years.
A $30,000 car depreciating 20% annually: after 5 years A = 30,000 × (0.80)⁵ = $9,830 — only 32.8% of original value. Exponential depreciation means steeper losses in early years.
⚡ Capacitor Discharge
Voltage decays as V = V₀e^(−t/RC). A 9V capacitor with RC = 2s: after 4 seconds V = 9 × e^(−2) = 9 × 0.1353 = 1.22V. This exponential decay appears throughout electronics engineering.
Voltage decays as V = V₀e^(−t/RC). A 9V capacitor with RC = 2s: after 4 seconds V = 9 × e^(−2) = 9 × 0.1353 = 1.22V. This exponential decay appears throughout electronics engineering.
Exponential Growth vs Linear Growth — The Difference
Linear growth adds the same amount each period (y = mx + b). Exponential growth multiplies by the same factor each period (y = A₀(1+r)ᵗ). The difference starts small but becomes staggering over time.
| Year | Linear (+100/yr) | Exponential (10%/yr) | Difference |
|---|---|---|---|
| 0 | 1,000 | 1,000 | 0 |
| 5 | 1,500 | 1,611 | +111 |
| 10 | 2,000 | 2,594 | +594 |
| 15 | 2,500 | 4,177 | +1,677 |
| 20 | 3,000 | 6,727 | +3,727 |
| 30 | 4,000 | 17,449 | +13,449 |
| 40 | 5,000 | 45,259 | +40,259 |
| 50 | 6,000 | 117,391 | +111,391 |
Starting amount: 1,000 | Linear: adds 100/year | Exponential: grows 10%/year
💡 Key insight: In early periods, linear and exponential growth look similar — the exponential advantage compounds slowly at first. But over long periods, exponential growth completely dominates. This is why long-term investing, population growth, and technological progress produce results that seem impossibly large. The exponential function grows faster than any polynomial — eventually.
Worked Examples
1. Calculate exponential growth: A = A₀(1+r)ᵗ
$5,000 at 7% for 20 years → A = 5,000 × (1+0.07)²⁰ = 5,000 × (1.07)²⁰ = 5,000 × 3.8697 = $19,348
$5,000 at 7% for 20 years → A = 5,000 × (1+0.07)²⁰ = 5,000 × (1.07)²⁰ = 5,000 × 3.8697 = $19,348
2. Find initial amount A₀ from final amount
A = $5,000, r = 7%, t = 20 yr → A₀ = A/(1+r)ᵗ = 5,000/(1.07)²⁰ = 5,000/3.8697 = $1,292.10
A = $5,000, r = 7%, t = 20 yr → A₀ = A/(1+r)ᵗ = 5,000/(1.07)²⁰ = 5,000/3.8697 = $1,292.10
3. Find growth rate from start and end values
100,000 → 140,000 in 8 years → r = (140,000/100,000)^(1/8) − 1 = (1.4)^0.125 − 1 = 1.0432 − 1 = 4.32% per year
100,000 → 140,000 in 8 years → r = (140,000/100,000)^(1/8) − 1 = (1.4)^0.125 − 1 = 1.0432 − 1 = 4.32% per year
4. Find time using logarithms
$1,000 → $5,000 at 8% → t = ln(5,000/1,000)/ln(1.08) = ln(5)/ln(1.08) = 1.6094/0.07696 = 20.9 years
$1,000 → $5,000 at 8% → t = ln(5,000/1,000)/ln(1.08) = ln(5)/ln(1.08) = 1.6094/0.07696 = 20.9 years
5. Continuous exponential growth: A = A₀eʳᵗ
A₀=1,000, r=5%, t=10 yr → A = 1,000 × e^(0.05×10) = 1,000 × e^0.5 = 1,000 × 1.6487 = $1,648.72
A₀=1,000, r=5%, t=10 yr → A = 1,000 × e^(0.05×10) = 1,000 × e^0.5 = 1,000 × 1.6487 = $1,648.72
6. Calculate half-life for exponential decay
r = 5% decay → t_h = ln(0.5)/ln(1−0.05) = ln(0.5)/ln(0.95) = −0.6931/(−0.05129) = 13.51 periods. Continuous: t_h = 0.693/0.05 = 13.86 periods
r = 5% decay → t_h = ln(0.5)/ln(1−0.05) = ln(0.5)/ln(0.95) = −0.6931/(−0.05129) = 13.51 periods. Continuous: t_h = 0.693/0.05 = 13.86 periods
7. Rule of 72 doubling time estimate
At 8% growth → Rule of 72: 72/8 = 9 periods (approximate). Exact: ln(2)/ln(1.08) = 0.6931/0.07696 = 9.006 periods. Rule of 72 error: 0.07%. Excellent approximation.
At 8% growth → Rule of 72: 72/8 = 9 periods (approximate). Exact: ln(2)/ln(1.08) = 0.6931/0.07696 = 9.006 periods. Rule of 72 error: 0.07%. Excellent approximation.
8. Population growth over time
P₀=1,000,000, r=2%/yr, t=50 yr → P = 1,000,000 × e^(0.02×50) = 1,000,000 × e^1 = 1,000,000 × 2.71828 = 2,718,282 people. Doubling time = 0.693/0.02 = 34.66 years.
P₀=1,000,000, r=2%/yr, t=50 yr → P = 1,000,000 × e^(0.02×50) = 1,000,000 × e^1 = 1,000,000 × 2.71828 = 2,718,282 people. Doubling time = 0.693/0.02 = 34.66 years.
9. Find decay rate from two data points
200g decays to 80g in 15 years → r = ln(A/A₀)/t = ln(80/200)/15 = ln(0.4)/15 = −0.9163/15 = −0.0611/yr = −6.11%/yr
200g decays to 80g in 15 years → r = ln(A/A₀)/t = ln(80/200)/15 = ln(0.4)/15 = −0.9163/15 = −0.0611/yr = −6.11%/yr
10. Compare continuous vs standard compounding
$1,000 at 5% for 10 years: Annual (1 compound/yr) = $1,628.89 · Monthly = $1,647.01 · Daily = $1,648.61 · Continuous = $1,648.72. Difference between annual and continuous: $19.83 (1.22% more).
$1,000 at 5% for 10 years: Annual (1 compound/yr) = $1,628.89 · Monthly = $1,647.01 · Daily = $1,648.61 · Continuous = $1,648.72. Difference between annual and continuous: $19.83 (1.22% more).
Frequently Asked Questions
What is the exponential growth formula?
The standard exponential growth formula is A = A₀(1+r)ᵗ, where A is the final amount, A₀ is the initial amount, r is the growth rate as a decimal (5% = 0.05), and t is the number of time periods. For continuous growth, use A = A₀eʳᵗ where e = 2.71828. Both formulas appear multiple times throughout this page with full worked examples.
What is the difference between exponential growth and decay?
Exponential growth uses A = A₀(1+r)ᵗ with positive r — the quantity increases each period. Exponential decay uses A = A₀(1−r)ᵗ — the same formula but r is subtracted, reducing the quantity each period. The quantity never reaches zero; it approaches asymptotically. Both growth and decay follow a curved path rather than a straight line.
How do you calculate exponential growth rate?
Rearrange A = A₀(1+r)ᵗ: divide both sides by A₀ to get (A/A₀) = (1+r)ᵗ, then raise both sides to 1/t: (A/A₀)^(1/t) = 1+r, so r = (A/A₀)^(1/t) − 1. For continuous growth: r = ln(A/A₀) / t. Enter your initial and final values into the calculator above and select "Find Rate r" mode.
What is the Rule of 72 for doubling time?
The Rule of 72 estimates doubling time: Doubling Time ≈ 72 ÷ Rate%. At 6% growth: 72/6 = 12 periods (exact: 11.90). At 9%: 72/9 = 8 periods (exact: 8.04). The Rule of 72 is remarkably accurate for rates between 3% and 12%. The exact formula is t_d = ln(2)/ln(1+r) ≈ 0.6931/ln(1+r).
What is continuous exponential growth?
Continuous exponential growth uses A = A₀eʳᵗ where e = 2.71828 (Euler's number). Unlike standard growth which compounds once per period, continuous growth compounds infinitely often — at every instant. It's the mathematical limit of (1+r/n)^(nt) as n approaches infinity. It models radioactive decay, bacterial growth, and continuous compounding in finance.
How do you find time in an exponential growth equation?
For standard growth A = A₀(1+r)ᵗ: take logarithm of both sides → ln(A) = ln(A₀) + t×ln(1+r) → t = ln(A/A₀) / ln(1+r). For continuous growth A = A₀eʳᵗ: → t = ln(A/A₀) / r. You need the initial amount A₀, final amount A, and the growth rate r. Use the "Find Time t" solver mode in the calculator above.
What is the difference between standard and continuous exponential growth?
Standard growth A = A₀(1+r)ᵗ compounds once per time period. Continuous growth A = A₀eʳᵗ compounds infinitely often. At the same nominal rate, continuous always yields slightly more. Example: $1,000 at 5% for 10 years: standard = $1,628.89, continuous = $1,648.72 — a difference of $19.83. The two formulas are related: eʳ = (1+r_effective), where r_effective = eʳ − 1.
How does exponential growth relate to compound interest?
Compound interest is exponential growth: A = P(1+r/n)^(nt) where P is principal, r is annual rate, n is compounding frequency, t is years. As n approaches infinity (continuous compounding), this formula converges exactly to A = Peʳᵗ — the continuous exponential formula. All compound interest — annual, monthly, daily — follows the exponential growth formula A = A₀(1+r)ᵗ.
What is exponential decay in real life?
Exponential decay occurs wherever a quantity decreases proportional to its current size. Real examples: radioactive decay (carbon-14 half-life 5,730 years), drug metabolism (ibuprofen half-life ~2 hours), car depreciation (15-20% per year), capacitor discharge (V = V₀e^(−t/RC)), Newton's Law of Cooling, and sound intensity with distance. All follow A = A₀(1−r)ᵗ or A = A₀e^(−rt).
How do you read an exponential growth graph?
An exponential growth graph curves upward, starting nearly flat and becoming dramatically steeper over time. The y-intercept equals A₀ (initial amount). The x-axis is time; the y-axis is quantity. A steeper curve indicates a higher growth rate. For decay, the curve drops steeply at first then flattens, approaching but never touching zero. Hover over the graph in this calculator to see exact (time, amount) values at any point.