Exponential Growth Calculator
Growth and decay, growth rate, doubling time and half-life
The exponential growth formula is x(t) = x₀ × (1 + r)t: multiply the starting value by the growth factor (1 + r) once for every period. A city of 85,000 people growing 2.5% a year reaches 85,000 × 1.02520 = 139,282 after 20 years. For exponential decay use a negative rate, and for continuous growth use x(t) = x₀ × ekt.
Last updated October 2026
Method: Discrete growth x(t) = x₀(1 + r)t and continuous growth x(t) = x₀ekt, rearranged with natural logarithms to solve for any one of the four quantities. The two rates are linked by k = ln(1 + r), doubling time and half-life equal ln 2 ÷ |k|, and the rate between two data points is (y₂ ÷ y₁)1/(t₂ − t₁) − 1.
Included: Final value, initial value, rate or time; growth or decay; discrete or continuous compounding; doubling time or half-life; a period-by-period table; and a two-point mode with the compound and simple average rates, the function y = a·bt and a projection.
Not included: Logistic (S-shaped) growth with a ceiling, rates that change over time, deposits or withdrawals along the way, curve fitting through three or more points, and statistical confidence ranges.
🚀 Set up your growth problem
🚀 Final value after 20 years
📋 Step-by-step
- 1. Formula. x(t) = x₀ × (1 + r)^t, with r negative for decay and t counted in years.
- 2. Growth factor. (1 + 0.025)^20 = 1.638616
- 3. Multiply. 85,000 × 1.638616 = 139,282.4
- 4. Doubling time. ln(2) ÷ |ln(1.025)| = 28.071 years (Rule of 70 shortcut: 70 ÷ 2.5 = 28)
📈 Year-by-year table
| Year | Value | Change | Total change |
|---|---|---|---|
| 0 | 85,000 | - | - |
| 1 | 87,125 | +2,125 | +2.5% |
| 2 | 89,303.13 | +2,178.13 | +5.06% |
| 3 | 91,535.7 | +2,232.58 | +7.69% |
| 4 | 93,824.1 | +2,288.39 | +10.38% |
| 5 | 96,169.7 | +2,345.6 | +13.14% |
| 6 | 98,573.94 | +2,404.24 | +15.97% |
| 7 | 101,038.29 | +2,464.35 | +18.87% |
| 8 | 103,564.25 | +2,525.96 | +21.84% |
| 9 | 106,153.35 | +2,589.11 | +24.89% |
| 10 | 108,807.19 | +2,653.83 | +28.01% |
| 11 | 111,527.37 | +2,720.18 | +31.21% |
| 12 | 114,315.55 | +2,788.18 | +34.49% |
| 13 | 117,173.44 | +2,857.89 | +37.85% |
| 14 | 120,102.77 | +2,929.34 | +41.3% |
| 15 | 123,105.34 | +3,002.57 | +44.83% |
| 16 | 126,182.98 | +3,077.63 | +48.45% |
| 17 | 129,337.55 | +3,154.57 | +52.16% |
| 18 | 132,570.99 | +3,233.44 | +55.97% |
| 19 | 135,885.27 | +3,314.27 | +59.87% |
| 20 | 139,282.4 | +3,397.13 | +63.86% |
Exponential growth formula, growth rate and decay explained
An exponential growth calculator takes three of the starting value, rate, time and ending value and finds the fourth. The same formula covers exponential decay, continuous growth with e, and measuring a growth rate from two observations.
The exponential growth formula
In exponential growth a quantity changes by the same percentage every period, so each step is larger than the last in absolute terms; linear growth adds the same amount instead. The standard forms raise a growth factor to the number of periods:
x(t) = x₀ × (1 + r)t (discrete: rate applied once per period) x(t) = x₀ × ek × t (continuous) k = ln(1 + r) r = ek − 1 doubling time = ln 2 ÷ k half-life = ln 2 ÷ |k| Here x₀ is the initial value, r is the growth rate per period written as a decimal (5% = 0.05), t is the number of periods and e ≈ 2.718281828 is the base of the natural logarithm. For decay the rate is negative: a 15% annual loss is r = −0.15, a growth factor of 0.85. The calculator takes a positive percentage plus a Growth or Decay switch, which prevents sign mistakes.
Worked example 1: a city growing 2.5% a year
A city has 85,000 residents and grows 2.5% a year. After 20 years:
x(20) = 85,000 × 1.02520 = 85,000 × 1.638616 = 139,282.40 The city gains 54,282 people, a 63.86% increase, at an unchanging rate: the first year adds 2,125 residents and the twentieth adds 3,397, because each 2.5% is taken from a bigger base. After 10 years the city has 108,807; after 15 it has 123,105. The doubling time is ln 2 ÷ ln 1.025 = 28.07 years, and the Rule of 70 (70 ÷ 2.5 = 28) lands within a month of it. Compounding the same 2.5% continuously instead of once a year would give 85,000 × e0.5 = 140,141, about 859 more people.
Worked example 2: a car losing 15% of its value a year
Exponential decay is the same formula with a negative rate. Suppose a $32,000 car loses 15% of its remaining value every year (an assumed rate for illustration). After six years:
x(6) = 32,000 × (1 − 0.15)6 = 32,000 × 0.377150 = $12,068.78 | Year | Value | Lost that year | Share of price left |
|---|---|---|---|
| 0 | $32,000.00 | - | 100% |
| 1 | $27,200.00 | $4,800.00 | 85% |
| 2 | $23,120.00 | $4,080.00 | 72.25% |
| 3 | $19,652.00 | $3,468.00 | 61.41% |
| 4 | $16,704.20 | $2,947.80 | 52.20% |
| 5 | $14,198.57 | $2,505.63 | 44.37% |
| 6 | $12,068.78 | $2,129.79 | 37.71% |
The car has lost $19,931.22, or 62.29% of its price. Each year's 15% comes out of a smaller value, so the dollar losses shrink from $4,800 in the first year to $2,129.79 in the sixth. The half-life is ln 2 ÷ (−ln 0.85) = 4.27 years whatever the price, and the value approaches zero without ever arriving.
The growth rate formula: the rate between two data points
When you know where a quantity started and ended but not the rate, solve the formula for r. Over n periods:
r = (end ÷ start)1/n − 1 k = ln(end ÷ start) ÷ n The U.S. Census Bureau counted 308,745,538 residents in 2010 and 331,449,281 in 2020. The total change is 331,449,281 ÷ 308,745,538 − 1 = 7.35%, and the average annual growth rate is (331,449,281 ÷ 308,745,538)1/10 − 1 = 0.7121% per year. Dividing the total by ten gives 0.7354%, slightly too high, because that simple average ignores compounding. At 0.7121% a year the population would double in 97.69 years, and the same pace would give 355,822,554 people in 2030 (an extrapolation, not a Census projection).
U.S. population growth rate by decade
Applying the same formula to every census since 1950 shows how the country's growth has slowed. The populations are the official census counts; every rate and doubling time was computed from them.
| Decade | Population at end | Total change | Average annual rate | Doubling time at that pace |
|---|---|---|---|---|
| 1950-1960 | 179,323,175 | 18.50% | 1.71% | 40.8 years |
| 1960-1970 | 203,211,926 | 13.32% | 1.26% | 55.4 years |
| 1970-1980 | 226,545,805 | 11.48% | 1.09% | 63.8 years |
| 1980-1990 | 248,709,873 | 9.78% | 0.94% | 74.3 years |
| 1990-2000 | 281,421,906 | 13.15% | 1.24% | 56.1 years |
| 2000-2010 | 308,745,538 | 9.71% | 0.93% | 74.8 years |
| 2010-2020 | 331,449,281 | 7.35% | 0.71% | 97.7 years |
The 1950 census counted 151,325,798 people, so over 70 years the population grew 119.03%, an average of 1.1264% a year (doubling time 61.89 years). The 1950s ran well above that average and the 2010s well below: one average rate summarizes a path without describing it.
Continuous growth and the number e
Discrete growth applies the rate once per period; continuous growth applies it in endlessly many tiny steps, and the limit of (1 + k/n)n as n grows is ek. That is why science texts write growth as x₀ekt. A continuous rate of 5% equals an effective rate of e0.05 − 1 = 5.1271% per period, and an effective rate of 2.5% equals a continuous rate of ln 1.025 = 2.4693%.
Worked example 3: a lab culture starts with 500 cells and grows at a continuous rate of k = 0.35 per hour. After 8 hours:
x(8) = 500 × e0.35 × 8 = 500 × e2.8 = 500 × 16.444647 = 8,222.32 The culture passes 1,006.88 cells at hour 2, 2,027.60 at hour 4 and 4,083.08 at hour 6, doubling every ln 2 ÷ 0.35 = 1.98 hours, an effective hourly rate of 41.91%. Treating the same 35% as a once-per-hour discrete rate would give only 500 × 1.358 = 5,516.20 cells, so check which kind of rate your textbook or data source means.
Doubling time and half-life at common rates
This table compares exact doubling times for discrete and continuous growth with the Rule of 70 shortcut, and adds the half-life for decay at the same percentage. Values are in periods, so years when the rate is annual.
| Rate per period | Doubling, discrete | Doubling, continuous | Rule of 70 | Half-life at this decay rate |
|---|---|---|---|---|
| 1% | 69.66 | 69.31 | 70.00 | 68.97 |
| 2% | 35.00 | 34.66 | 35.00 | 34.31 |
| 3% | 23.45 | 23.10 | 23.33 | 22.76 |
| 5% | 14.21 | 13.86 | 14.00 | 13.51 |
| 7% | 10.24 | 9.90 | 10.00 | 9.55 |
| 10% | 7.27 | 6.93 | 7.00 | 6.58 |
| 15% | 4.96 | 4.62 | 4.67 | 4.27 |
| 20% | 3.80 | 3.47 | 3.50 | 3.11 |
| 25% | 3.11 | 2.77 | 2.80 | 2.41 |
| 50% | 1.71 | 1.39 | 1.40 | 1.00 |
The Rule of 70 tracks the continuous column at every rate and the discrete column up to about 10%, then drifts low. Decay also halves faster than growth doubles: losing 50% a period halves a value in one period, while gaining 50% takes 1.71 periods to double it.
How to use this calculator
Choose Growth / decay to solve the formula or Two data points to measure a rate. Pick what to solve for, enter the other three values, then choose Growth or Decay and discrete or continuous compounding. The time unit only labels results, but the rate and the time must use the same unit.
The result card shows the answer, the formula with your numbers, the growth factor, the total change, the doubling time or half-life and the equivalent rate in the other model; below it come the algebra and a period-by-period table. Solving for time, 5,000 growing to 8,000 at 6% a year takes ln 1.6 ÷ ln 1.06 = 8.07 years. Solving for the rate, 500 becoming 2,000 in 8 years needs 18.92% a year. Solving for the initial value, reaching 50,000 in 18 years at 7% requires starting with 14,793.20.
Who this calculator is for
Algebra and precalculus students check exponential function homework, including y = a·bx through two points: (1, 6) and (3, 54) give y = 2·3x. Science students model cultures and decay, analysts measure the compound growth of users or sales between two dates, and planners run what-if projections. Anyone reading a claim such as "doubling every three years" can convert it into a rate (25.99% a year) and judge whether it is plausible.
Key terms
- Initial value (x₀): the quantity at time zero.
- Growth factor: 1 + r per period; above 1 for growth, between 0 and 1 for decay.
- Growth rate (r): the percentage change per period, written as a decimal in the formula.
- Continuous rate (k): the rate in x₀ekt; k = ln(1 + r).
- Doubling time: the time a growing quantity needs to double, ln 2 ÷ k.
- Half-life: the time a decaying quantity needs to halve, ln 2 ÷ |k|.
- Average annual growth rate: the constant yearly rate that links a start and an end value, the same idea as CAGR.
- Exponential function: y = a·bx with a > 0, b > 0 and b ≠ 1.
What changes the result
- The rate, compounded. Raising the city's rate from 2.5% to 3% adds about 14,237 residents after 20 years (153,519 instead of 139,282).
- The time. Doubling the horizon squares the growth factor: 40 years at 2.5% multiply the start by 2.685064, giving 228,230 rather than twice 139,282.
- The period of the rate. 1% a month compounds to 12.68% a year, and 12% a year is only 0.9489% a month.
- Discrete or continuous. The same number means a larger effective rate when it is continuous, as the lab culture shows.
Tips and limitations
Estimate a rate from as long a stretch of real data as you trust, and sanity-check answers with the doubling time: 7% doubles about every 10 years, so 30 years means roughly three doublings (the exact factor is 7.61). Values must be positive; zero cannot grow by a percentage.
Real growth rarely stays exponential. Populations and markets slow as they near a ceiling (logistic growth), rates shift, and regular deposits add a term this formula leaves out, so for savings with contributions use a compound interest model.
Related calculators
- For savings with regular deposits and compounding frequencies, use the Compound Interest Calculator.
- For the annualized return of an investment between two values, use the CAGR Calculator.
- For radioactive isotopes and decay constants, use the Half-Life Calculator.
- For the quick doubling-time estimate, use the Rule of 72 Calculator.
- For the natural logarithms used to solve for time, use the Logarithm Calculator.
- For a one-step change between two numbers, use the Percentage Change Calculator.
Sources
The growth and decay formulas are exact mathematics and need no external figures; the population example uses official census counts.
- U.S. Census Bureau: Historical Population Change Data (1910-2020), resident population of the United States by census year.
- National Institute of Standards and Technology (NIST): Digital Library of Mathematical Functions, §4.2: definitions of the exponential function and the natural logarithm.
- NIST: Digital Library of Mathematical Functions, §4.4: special values and limits, including (1 + z/n)n → ez (eq. 4.4.17).
- All other figures on this page were computed directly from the formulas above and checked against the calculator.
⚠️ Common mistakes & edge cases
Writing the percent instead of the decimal
By hand, 5% growth means a factor of 1.05, not 1 + 5 = 6. The calculator takes the percentage directly, so type 5, not 0.05.
Mixing time units
A monthly rate with a time in years makes the exponent twelve times too small. Convert first: 2% a month is 26.82% a year.
Averaging percentage changes
A gain of 50% followed by a loss of 50% averages to 0%, yet the value ends at 75% of where it began. Use the compound rate from the two data points mode whenever you summarize several periods.
Using a continuous rate in the discrete formula
If a source gives k = 0.35 per hour, plugging 35% into (1 + r)t undercounts badly: 5,516 cells after 8 hours instead of 8,222. Switch the calculator to Continuous when the rate is continuous.
Decay rates of 100% or more
With discrete compounding a 100% loss erases the value in one period, so the calculator rejects it. Continuous decay above 100% is valid: k = 1.5 multiplies the value by e−1.5 ≈ 0.22 each period.
Projecting far beyond the data
A rate from one decade says little about the next five: the 1950s population pace doubles in 40.8 years, the 2010s pace in 97.7.
❓ Frequently asked questions
What is the exponential growth formula?
The exponential growth formula is x(t) = x0 × (1 + r)^t. The starting value x0 is multiplied by the growth factor (1 + r) once per period, where r is the rate as a decimal. A city of 85,000 people growing 2.5% a year reaches 85,000 × 1.025^20 = 139,282.40 after 20 years. For continuous growth use x(t) = x0 × e^(kt).
How do you calculate exponential decay?
Use the same formula with the rate subtracted from 1: x(t) = x0 × (1 − r)^t. A quantity of 1,200 that shrinks 8% per period for 10 periods ends at 1,200 × 0.92^10 = 521.27. Its half-life is ln 2 ÷ (−ln 0.92) = 8.31 periods. In the calculator, enter 8 as the rate and switch to Decay.
How do I calculate the growth rate between two numbers?
Divide the end value by the start value, raise the result to the power 1 ÷ n, where n is the number of periods between them, and subtract 1. Going from 40 to 65 over 6 years gives (65 ÷ 40)^(1/6) − 1 = 8.43% per year, and six years of 8.43% compound to the total increase of 62.5%.
Why is the average growth rate not the total percent change divided by the years?
Because growth compounds. From 40 to 65 is a 62.5% total change, and 62.5 ÷ 6 = 10.42%, but 10.42% a year for six years would carry 40 to about 72, not 65. Only the compound rate of 8.43% reproduces the actual ending value. The simple average overstates growth, and the gap widens as the total change grows.
What is the difference between exponential and linear growth?
Linear growth adds the same amount each period; exponential growth multiplies by the same factor each period. Starting from 1,000, adding 100 a year gives 2,000 after 10 years, while growing 10% a year gives 1,000 × 1.1^10 = 2,593.74. The paths start out close, but the exponential one pulls away faster every year.
How long does it take for something to double at a given growth rate?
The exact doubling time is ln 2 ÷ ln(1 + r) periods: at 7% a year, ln 2 ÷ ln 1.07 = 10.24 years. The Rule of 70 gives a quick estimate, 70 ÷ 7 = 10 years. It comes from continuous growth, where the doubling time is exactly ln 2 ÷ k, about 69.3 divided by the rate in percent.
When should I use e^(kt) instead of (1 + r)^t?
Use x0 × e^(kt) when the rate is described as continuous, as is common in biology, physics and calculus. Use x0 × (1 + r)^t when the rate is applied once per period. The two are linked by k = ln(1 + r): a continuous rate of 5% equals an effective rate of e^0.05 − 1 = 5.1271% per period.
How do I convert a monthly growth rate to an annual rate?
Compound it rather than multiplying by 12: annual rate = (1 + monthly rate)^12 − 1. A 2% monthly growth rate is 1.02^12 − 1 = 26.82% a year, not 24%. In the other direction, 12% a year is 1.12^(1/12) − 1 = 0.9489% a month.
How do I find an exponential function from two points?
Write the function as y = a × b^x. The base is b = (y2 ÷ y1)^(1 ÷ (x2 − x1)) and then a = y1 ÷ b^x1. For the points (1, 6) and (3, 54), b = (54 ÷ 6)^(1/2) = 3 and a = 6 ÷ 3 = 2, so y = 2 × 3^x and the value at x = 5 is 486. The two data points mode does this for any pair of positive values.
Can exponential growth continue forever?
Not in the physical world. Populations run out of space, markets saturate and outbreaks run out of susceptible people, so real curves bend into an S-shape called logistic growth. Exponential models describe the early phase well and are exact for compound interest without withdrawals, but long projections are what-if scenarios, not forecasts.
💡 Good to know
A constant doubling time is the fingerprint of exponential change
The example city doubles every 28.07 years whether it has 85,000 residents or 170,000. If a data series keeps taking about the same time to double (or halve), an exponential model fits; if that time keeps stretching, growth is slowing.
Percentage changes multiply, they do not add
A rise of 10% followed by a fall of 10% leaves 1.1 × 0.9 = 0.99 of the original, a 1% loss. Growth over several periods combines by multiplying factors, which is what the exponent does.
Rule of 70 or Rule of 72
Both approximate ln 2 ÷ rate. Seventy fits continuous growth best (100 × ln 2 ≈ 69.3); 72 divides evenly by more numbers and suits yearly compounding at typical interest rates.