What this calculator does
See how big something gets, or how small it shrinks, when it changes by the same percent over and over. That kind of change is called exponential. Each period it grows or shrinks by the same percentage, not by the same amount. A population that grows 5% a year follows this pattern. So does a medicine that clears from the body at a steady share per hour, and savings that earn compound interest. The bigger the amount gets, the faster it grows. In decay, the smaller it gets, the more slowly it falls.
This exponential growth and decay calculator works with either of two models. One is a percent change per period, P = P₀(1 ± r)ᵗ. The other is smooth, nonstop (continuous) change, P = P₀e^(kt). Give it the starting amount, the rate and the time to get the final amount. Or give it a target to find how long it takes. Or give it a start and an end to find the rate. Every answer includes the doubling time or half-life and the matching rate in the other model. You also get a period-by-period table and a chart.
Where it's used
Why this math matters outside the classroom:
- Population planning. A planner projects that a town of 1,000 growing 5% a year will have about 1,629 residents after 10 years.
- Medicine clearance. A drug that leaves the body at 20% per hour drops to 0.8⁴ ≈ 41% of the dose after 4 hours.
- Savings growth. A saver sees that $1,000 earning 5% a year takes about 14.2 years to double to $2,000.
- Car depreciation. A $30,000 car losing 15% of its value each year is worth about 30,000 × 0.85³ ≈ $18,424 after 3 years.
- Lab cultures. Biologists find that a culture growing from 1,000 to 2,000 cells in 10 hours has a continuous growth rate of about 6.93% per hour.
Grade level and standards: Grades 9–11; Common Core State Standards for Mathematics HSF.LE.A.1.c, HSF.LE.A.4, HSF.LE.B.5. © Copyright 2010. National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved.
How to use it
- Choose what to solve for: the final amount, the time to reach a target, or the rate.
- Choose growth or decay. Then choose whether the rate is a percent per period or a continuous rate.
- Enter the starting amount and the other values you know. The time can be in any unit, as long as the rate is per that same unit.
- Read the answer at the top. Below it are the doubling time or half-life, the total change and the table of amounts over time.
What the inputs mean
- Rate type
- Percent per period suits things that change in steps, such as yearly population counts or yearly interest. Continuous suits things that change smoothly every instant, such as radioactive decay or cooling.
- Rate per period
- Enter 5 for 5%. A decay rate per period must be below 100%. A continuous decay rate can be any positive value.
- Number of periods (t)
- Fractions are fine: 2.5 periods means two and a half periods.
Formula and method
Percent per period: P = P₀ × (1 + r)ᵗ for growth, P₀ × (1 − r)ᵗ for decay Continuous: P = P₀ × e^(kt), with k > 0 for growth and k < 0 for decay Time to a target: t = ln(P ÷ P₀) ÷ ln(b) = ln(P ÷ P₀) ÷ k Rate: b = (P ÷ P₀)^(1/t), r = b − 1; continuous k = ln(P ÷ P₀) ÷ t Doubling time or half-life = ln 2 ÷ |k|, where k = ln(b)
P₀ is the starting amount, P is the amount after t periods, and r is the rate per period as a decimal (0.05 for 5%). k is the continuous rate. e is a special math constant, about 2.718, and ln is the natural log (the log based on e). |k| means k without its minus sign. In the percent model, the amount is multiplied by the same factor every period. That factor is b = 1 + r, or b = 1 − r for decay. After t periods, it's been multiplied by b a total of t times, which is bᵗ.
The continuous model adds the change at every instant, and that's where the number e comes in. A continuous rate k and a per-period factor b draw the same curve when b = e^k. So the calculator shows each model's match in the other.
To find the time or the rate, the formula is turned around with logarithms (logs). Dividing the target by the start gives the overall factor. The natural log of that factor, divided by the log of the per-period factor, counts how many periods it takes.
The doubling time (or half-life) doesn't depend on the starting amount. It's always ln 2 ≈ 0.6931 divided by the continuous rate. That's where the rule of 70 comes from: at 5% continuous growth, 69.3 ÷ 5 ≈ 13.9 periods.
Worked example
A town of 1,000 people grows by 5% a year. How big is it after 10 years, and when does it reach 2,000?
- Growth factor: b = 1 + 0.05 = 1.05 per year.
- After 10 years: P = 1,000 × 1.05¹⁰ ≈ 1,628.8946 people.
- Time to 2,000: t = ln(2,000 ÷ 1,000) ÷ ln(1.05) = 0.6931 ÷ 0.0488 ≈ 14.2067 years. This is also the doubling time.
- Working backward: reaching 2,000 in exactly 10 years would need b = 2^(1/10) ≈ 1.0718, a growth rate of about 7.1773% a year.
- With 5% continuous growth instead, the town would reach 1,000 × e^0.5 ≈ 1,648.7213 after 10 years.
Result: The town reaches about 1,628.8946 people (call it 1,629) after 10 years. It doubles to 2,000 after about 14.2067 years.
Understanding the result
A doubling time or half-life gives you a feel for how fast things change. Something growing 5% a year doubles roughly every 14 years. So it grows four times as big in about 28 years, and eight times as big in about 43.
Small differences in rate grow into big differences over time. Over 50 years, 5% growth multiplies an amount by about 11.5, while 7% multiplies it by about 29.5.
A negative rate from the 'rate' option means decay. Its size is the percentage lost per period.
Assumptions and limitations
- The rate stays the same for the whole time span.
- The percent model applies the change once per period. For a stated rate compounded several times per period, use the compound interest calculator.
- Time and rate use the same unit (both per year, or both per hour).
- Real populations slow down in the end as resources run out. A logistic model, which levels off, fits those better than endless exponential growth.
- Results aren't rounded to whole units, so a population or bacteria count may come out as a fraction.
- Very large rates or very long times can go beyond what a computer can hold, and the calculator tells you so.
Frequently asked questions
What is the formula for exponential growth?
For a percent rate per period, P = P₀(1 + r)ᵗ. P₀ is the starting amount, r is the rate as a decimal and t is the number of periods. For continuous growth, P = P₀e^(kt). For decay, use 1 − r or a negative k.
How do you find the doubling time?
Divide ln 2 by the natural log of the growth factor: t = ln 2 ÷ ln(1 + r). At 5% per period that's 0.6931 ÷ 0.0488 ≈ 14.2 periods. For a continuous rate, it's ln 2 ÷ k.
What is the difference between exponential and linear growth?
Linear growth adds the same amount each period, while exponential growth multiplies by the same factor. Starting at 1,000, adding 50 a year gives 1,500 after 10 years. Growing 5% a year gives about 1,629, and the gap keeps widening.
How is a percent rate converted to a continuous rate?
Take the natural log of the growth factor: k = ln(1 + r). A 5% per-period rate equals a continuous rate of ln 1.05 ≈ 4.879%. To go the other way, use r = e^k − 1.
How do I model a half-life with this calculator?
Choose decay and the continuous model. Then enter the rate k = ln 2 ÷ half-life as a percent. For a half-life of 8 days, k = 0.6931 ÷ 8 ≈ 8.664% per day, and the half-life line will read 8.
How this calculator is checked
Method: The discrete exponential model P₀(1 ± r)ᵗ and the continuous model P₀e^(kt), solved for time and rate with natural logarithms; doubling time and half-life ln 2/|k|. The calculation is covered by automated tests with known reference values and edge cases. Last updated 2026-10-18. Found a problem? Tell us.
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