Exponential Growth Calculator
Choose discrete growth, A = P(1 + r)^t, or continuous growth, A = Pe^(rt), growth or decay, and solve for any one of the final amount, initial amount, rate or time. You also get the doubling time or half-life, a comparison with the rules of 72, 70 and 69.3, the same rate written both ways, a table of values and a graph. Free, no sign-up.
Only used for labels, for example years, days or hours.
Results
1,628.8946
Final amount (A)
1.05
Factor per periods
+62.89%
Total change
5% per periods
As a discrete rate
4.879% per periods
As a continuous rate
14.2067 periods
Doubling time
The working
Rearrange the formula for the unknown.
A = P × (1 + r)^t
Substitute the numbers.
A = 1,000 × (1 + 0.05)^10 = 1,000 × 1.628895 = 1,628.8946
The growth factor for one period is 1.05.
1 + 0.05 = 1.05
Rule of 72, 70 and 69.3
The rules estimate the doubling or halving time as a constant divided by the rate in percent. Each is an approximation; the exact time is ln 2 divided by ln of the factor.
| Method | Time (periods) | Difference from exact |
|---|---|---|
| Exact | 14.2067 | — |
| Rule of 72: 72 / rate | 14.4 | +1.36% |
| Rule of 70: 70 / rate | 14 | -1.45% |
| Rule of 69.3: 69.3 / rate | 13.86 | -2.44% |
Table of values
| Time (periods) | Amount | Change from start |
|---|---|---|
| 0 | 1,000 | 0% |
| 1 | 1,050 | +5% |
| 2 | 1,102.5 | +10.25% |
| 3 | 1,157.625 | +15.76% |
| 4 | 1,215.5063 | +21.55% |
| 5 | 1,276.2816 | +27.63% |
| 6 | 1,340.0956 | +34.01% |
| 7 | 1,407.1004 | +40.71% |
| 8 | 1,477.4554 | +47.75% |
| 9 | 1,551.3282 | +55.13% |
| 10 | 1,628.8946 | +62.89% |
Graph
- A(t)
How to Use the Exponential Growth Calculator
- 1. Pick the model. Use discrete when the quantity grows by a fixed percentage once per period; use continuous when it grows by e raised to a rate times time.
- 2. Pick growth or decay and what to solve for. The field for the unknown is hidden. Enter the rate as a positive percentage: 5 means 5%, and the Growth or Decay choice sets its sign.
- 3. Fill in the other three values. The time unit is only a label, so years, days or hours all work.
- 4. Read the results. The solved value comes first, followed by the growth factor, the doubling time or half-life, the rules of thumb, a table of values and a graph.
The Two Models
| Model | Formula | Rate means | Doubling time |
|---|---|---|---|
| Discrete | A = P(1 + r)^t | r is the fractional change in each period; use 1 − r for decay | ln 2 / ln(1 + r) |
| Continuous | A = Pe^(rt) | r is the instantaneous rate; use −r for decay | ln 2 / r |
P is the initial amount, A the amount after time t, and r the rate. The two models describe the same curve when the rates are linked: a discrete rate r equals a continuous rate k = ln(1 + r), and k equals a discrete rate e^k − 1. Compounding more and more often at the same nominal rate gives the continuous model in the limit. The calculator shows your rate in both forms.
Worked Examples
| Problem | Calculation | Result |
|---|---|---|
| 1,000 growing 5% per period for 10 periods | 1000 × 1.05¹⁰ | 1,628.8946 |
| The same, but continuously | 1000 × e^(0.05 × 10) | 1,648.7213 |
| 1,000 decaying 10% per period for 3 periods | 1000 × 0.9³ | 729 |
| Time for 1,000 to reach 2,000 at 8% per period | ln 2 / ln 1.08 | 9.0065 periods |
| Rate for 1,500 to reach 2,400 in 8 periods | (2400 / 1500)^(1/8) − 1 | 6.0511% per period |
To solve for time or rate, rearrange with logarithms or roots: t = ln(A / P) / ln(1 + r) and r = (A / P)^(1/t) − 1. The calculator writes the rearranged formula and your numbers on separate lines.
Doubling Time, Half-Life and the Rule of 72
With growth, the doubling time is how long it takes the amount to double; with decay, the half-life is how long it takes to halve. Both depend only on the rate, not on the starting amount. The rule of 72 estimates the doubling time as 72 divided by the rate in percent. It is popular because 72 divides evenly by 2, 3, 4, 6, 8, 9 and 12, and it is most accurate for discrete rates near 8%, roughly 6% to 10%. The rule of 70 suits lower rates, such as inflation near 2%, and 69.3 is the natural choice for continuous growth because ln 2 ≈ 0.693.
| Case | Exact | Rule of 72 | Rule of 70 | Rule of 69.3 |
|---|---|---|---|---|
| Discrete, 8% | 9.0065 | 9 (−0.07%) | 8.75 (−2.85%) | 8.6625 (−3.82%) |
| Discrete, 5% | 14.2067 | 14.4 (+1.36%) | 14 (−1.45%) | 13.86 (−2.44%) |
| Continuous, 5% | 13.8629 | 14.4 (+3.87%) | 14 (+0.99%) | 13.86 (−0.02%) |
Treat the rules as mental arithmetic
The rules are quick estimates, not substitutes for the logarithm formula. They drift at high rates, and for decay they are rougher still, so use the exact time when it matters.
Common Mistakes
- Typing 0.05 for 5%. The rate field takes a percentage, so enter 5. Entering 0.05 means 0.05%.
- Using 1 + r for decay. A decay of 10% per period multiplies by 0.9, not 1.1. Choose Decay and the calculator uses 1 − r.
- Mixing the units of rate and time. A rate of 5% per year needs time in years; 5% per month needs months.
- Treating discrete and continuous rates as equal. 5% discrete is about 4.879% continuous, so the two models give different amounts for the same number.
- Expecting a negative or zero starting amount to work. The model needs a positive initial and final amount, because logarithms are only defined for positive numbers.
Frequently Asked Questions
- What is the formula for exponential growth?
- For growth by a fixed percentage each period it is A = P(1 + r)^t, where P is the initial amount, r the rate per period as a decimal, and t the number of periods. For continuous growth it is A = Pe^(rt). Decay uses a negative rate, so a 10% decay per period multiplies by 0.9.
- How do I calculate doubling time?
- For discrete growth the exact doubling time is ln 2 / ln(1 + r); for continuous growth it is ln 2 / r. At 8% per period that is 9.0065 periods. The rule of 72 gives a close estimate: 72 divided by the rate in percent.
- How accurate is the rule of 72?
- It is most accurate for rates around 8%, and good from about 6% to 10%. At 5% it is 1.4% too high for discrete growth. For continuous growth the rule of 69.3 is nearly exact, and the rule of 70 is better than 72 at low rates.
- What is the difference between discrete and continuous growth?
- Discrete growth applies the rate once per period, so 5% per year multiplies by 1.05 each year. Continuous growth applies it at every instant, which is the limit of compounding ever more often, and multiplies by e^0.05 ≈ 1.0513 each year. A discrete rate r is equal to a continuous rate ln(1 + r).
- How do I find the growth rate from two values?
- Choose Rate as the unknown and enter the initial amount, final amount and time. For discrete growth r = (A / P)^(1/t) − 1; for example growing from 1,500 to 2,400 in 8 periods is 6.0511% per period. A negative result means the amount is falling.
- How do I calculate half-life?
- Half-life is the time for the amount to fall to half: ln 2 divided by the size of the continuous decay rate, or ln 2 / −ln(1 − r) for a discrete decay of r per period. Choose Decay and enter the rate to see it.
- Can I use this for compound interest or population growth?
- Yes: both are exponential growth with a fixed rate per period. For interest with regular deposits, or with compounding several times per period, the Compound Interest Calculator is the closer fit. Real populations eventually stop growing exponentially, so use the result as a model rather than a forecast.