Math
Exponential Growth Calculator
Model anything that grows by the same percentage each period: populations, investments, bacteria, or views.
Pick the periodic model y = a(1 + r)^t or the continuous model y = a·e^(kt), solve for whichever variable is missing, and read off the doubling time, a table of values, and a graph.
The value at t = 0: population, balance, bacteria count, views.
Enter 5 for 5% growth each period. A negative rate models decay.
How many periods elapse. Must use the same unit as the rate.
Only labels the answer. The rate must be per one of these units.
Try an example
Result
Solved value
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- Final amount
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- Growth factor per period
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- Rate per period (%)
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- Continuous rate k
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- Doubling time
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- Equation
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Student quick launch
Grade planning, algebra checks, and formulas students reach for most.
Study path
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Follow these when you want the formula behind the answer, a short lesson, or nearby tools in the same topic.
What exponential growth means
Linear growth adds the same amount each period. Exponential growth multiplies by the same factor each period, so the increase itself keeps getting bigger. A 5% growth rate means each period's amount is 1.05 times the previous one.
What each symbol means
| Symbol | Meaning | Example |
|---|---|---|
| a | Starting amount at t = 0 | 1,000 people |
| r | Growth rate per period as a decimal | 5% → 0.05 |
| 1 + r | Growth factor: what you multiply by each period | 1.05 |
| k | Continuous rate; e^k is the growth factor | 0.0488 for 5% per period |
| t | Elapsed time in periods | 10 years |
| y | Amount after t periods | 1,628.89 |
Rearranged formulas the calculator uses
Periodic vs continuous growth
Both models describe the same shape of curve. The periodic model compounds once per period; the continuous model compounds at every instant. They are linked by 1 + r = e^k, so k = ln(1 + r). A 5% periodic rate equals a continuous rate of about 0.0488, and a continuous rate of 0.05 equals a periodic rate of about 5.13%.
How the rate changes the curve
Small differences in the rate compound into large differences over time. Starting from 1,000, here is the amount after 10 and 25 periods at four common rates, next to what straight-line growth (adding the same amount every period) would give.
| Rate per period | Growth factor | After 10 periods | Linear after 10 | After 25 periods | Doubling time |
|---|---|---|---|---|---|
| 1% | 1.01 | 1,104.62 | 1,100 | 1,282.43 | 69.66 periods |
| 3% | 1.03 | 1,343.92 | 1,300 | 2,093.78 | 23.45 periods |
| 5% | 1.05 | 1,628.89 | 1,500 | 3,386.35 | 14.21 periods |
| 10% | 1.10 | 2,593.74 | 2,000 | 10,834.71 | 7.27 periods |
Projecting to a calendar year
To project a Every member of the group you actually care about, not just the ones measured. from one year to another, set t to the number of years between them. A city of 10,000 growing 5% per year in 2019 reaches 10,000 × 1.05^11 ≈ 17,103 in 2030. To look backward, use a negative t: the same city had 10,000 × 1.05^(-19) ≈ 3,957 people in 2000 if the rate had held.
How to use this calculator
- Choose the model: percent per period, or continuous e^(kt).
- Choose what to solve for. The field for that variable disappears.
- Enter the other three values. The rate is a percent (5 for 5%) in the periodic model or a decimal k in the continuous model.
- Pick the time unit that matches the rate (a yearly rate needs t in years).
- Read the solved value, the growth factor, doubling time, table of values, and the step-by-step work.
How to read the answer
- Growth factor above 1 means growth; below 1 means decay; exactly 1 means no change.
- Doubling time is the same no matter where you start: it depends only on the rate.
- Percent change over the whole time is (y - a) / a × 100, which is larger than r × t because of compounding.
- The table shows the amount at evenly spaced times so you can see the curve steepen.
Common mistakes
- Entering 0.05 in the percent field. That means 0.05%, not 5%.
- Mixing units: a monthly rate with time in years. Convert first or change the time unit.
- Multiplying r by t instead of raising the growth factor to the power t.
- Using the continuous formula with a percent rate: e^(5t) explodes. Use k = 0.05, or switch to the percent model.
Worked examples
Population growth
1,000 people growing 5% per year for 10 years
Solved value
1,628.894627
Time to double
How long until 1,000 becomes 2,000 at 5% per year?
Solved value
14.206699
Find the growth rate
500 subscribers became 800 in 6 months
Solved value
8.148375
Continuous growth
$2,000 at a continuous rate of 0.03 for 8 years
Solved value
2,542.498301
Work backward to the start
What starting amount reaches 5,000 after 12 years at 7%?
Solved value
2,220.059796
Negative rate (decay)
800 shrinking 15% per year for 5 years
Solved value
354.96425
Zero rate
No growth: the amount never changes
Solved value
250
Project a city to 2030
10,000 people in 2019 growing 5% per year: t = 11
Solved value
17,103.393581
Look backward with negative time
The same city 19 years earlier (t = -19)
Solved value
3,957.33957
Time to triple
How long until 10,000 becomes 30,000 at 5% per year?
Solved value
22.517085
Continuous decay of caffeine
95 mg with k = -0.1155 per hour after 10 hours
Solved value
29.930466
Impossible: time with zero rate
Nothing can reach a different value when r = 0
Solved value
—
Frequently asked questions
What is the exponential growth formula?+
y = a(1 + r)^t, where a is the starting amount, r is the growth rate per period as a decimal, and t is the number of periods. The continuous version is y = a·e^(kt).
How do I find the growth rate from two values?+
Divide the final amount by the starting amount, take the t-th root, and subtract 1: r = (y / a)^(1/t) - 1. Choose Solve for rate and the calculator does this with steps.
How do I calculate doubling time?+
Doubling time = ln 2 / ln(1 + r) for a periodic rate, or ln 2 / k for a continuous rate. At 5% per year it is about 14.21 years. The Rule of 70 (70 / 5 = 14) is a quick estimate.
What is the difference between exponential growth and exponential decay?+
Growth has a growth factor above 1 (positive rate) and the amount increases. Decay has a factor between 0 and 1 (negative rate) and the amount decreases toward zero. This calculator handles both; the decay calculator uses half-life language.
Is compound interest exponential growth?+
Yes. A balance earning a fixed percentage each period follows y = a(1 + r)^t exactly, with r the periodic interest rate.
Why is the continuous rate k smaller than the percent rate?+
Because continuous compounding applies growth at every instant, a slightly smaller rate reaches the same result. They are linked by 1 + r = e^k, so k = ln(1.05) ≈ 0.0488 for 5%.
Can time be negative?+
Yes. A negative t projects backward and tells you what the amount was before the starting point, assuming the same rate held.
What is the difference between exponential and linear growth?+
Linear growth adds the same amount every period, so the graph is a straight line. Exponential growth multiplies by the same factor every period, so each increase is bigger than the last and the graph curves upward. At 5% from 1,000, linear growth reaches 1,500 after 10 periods; exponential growth reaches 1,628.89.
What happens when t = 0?+
Any growth factor raised to the power 0 is 1, so y = a. The amount has not had time to change, which is also why the calculator cannot solve for a rate when t = 0.
How do I use a yearly rate with time in months?+
Either convert the time to years (18 months = 1.5 years) or convert the rate: a 12% yearly rate compounded monthly is 1% per month, and a continuous yearly k becomes k/12 per month. Whatever you choose, the rate and t must use the same unit.
How do I project a population to a specific year?+
Set t to the gap between the years. From 2019 to 2030 is t = 11, so 10,000 at 5% becomes 10,000 × 1.05^11 ≈ 17,103. The table of values in the panel lists the in-between years.
About this calculator
- Written by
- mathcheck editorial team
- Last reviewed
- September 4, 2026
Method
- Uses the values entered by the user and stable formulas documented on the page.
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Last updated: September 4, 2026