Exponential Growth: worked example
| Starting value | 1,000 |
|---|---|
| Growth rate per period | 8.0% |
| Number of periods | 10 |
| Ending value | 2,158.9 |
| Periods to double | 9 |
| Total growth | 115.9% |
Exponential Growth: formula
Enter the starting value, the growth rate per period as a percentage, and the number of periods. The ending value equals the start times one plus the rate, raised to the number of periods. Use a negative rate for decay.
The period can be a day, a year or any repeating step, as long as the rate matches it. The doubling time is computed from the rate.
Exponential Growth: changing Growth rate per period
| Growth rate per period | Ending value | Periods to double | Total growth |
|---|---|---|---|
| 4.0% | 1,480.2 | 17.7 | 48.0% |
| 6.0% | 1,790.8 | 11.9 | 79.1% |
| 8.0% (example) | 2,158.9 | 9 | 115.9% |
| 10.0% | 2,593.7 | 7.3 | 159.4% |
| 12.0% | 3,105.8 | 6.1 | 210.6% |
Exponential Growth: compare input ranges
| Input | Tested from | To | Ending value: low input | Ending value: high input | Swing |
|---|---|---|---|---|---|
| Growth rate per period | 6.0% | 10.0% | 1,790.8 | 2,593.7 | 802.9 (37%) |
| Starting value | 900 | 1,100 | 1,943 | 2,374.8 | 431.8 (20%) |
| Number of periods | 9 | 11 | 1,999 | 2,331.6 | 332.6 (15%) |
Exponential Growth: individual input comparisons
| Input | Value | Ending value | Periods to double | Total growth |
|---|---|---|---|---|
| Starting value | 800 | 1,727.1 | 9 | 115.9% |
| 900 | 1,943 | 9 | 115.9% | |
| 1,000 | 2,158.9 | 9 | 115.9% | |
| 1,100 | 2,374.8 | 9 | 115.9% | |
| 1,200 | 2,590.7 | 9 | 115.9% | |
| Growth rate per period | 4.0% | 1,480.2 | 17.7 | 48.0% |
| 6.0% | 1,790.8 | 11.9 | 79.1% | |
| 8.0% | 2,158.9 | 9 | 115.9% | |
| 10.0% | 2,593.7 | 7.3 | 159.4% | |
| 12.0% | 3,105.8 | 6.1 | 210.6% | |
| Number of periods | 8 | 1,850.9 | 9 | 85.1% |
| 9 | 1,999 | 9 | 99.9% | |
| 10 | 2,158.9 | 9 | 115.9% | |
| 11 | 2,331.6 | 9 | 133.2% | |
| 12 | 2,518.2 | 9 | 151.8% |
Exponential Growth: combined input changes
| Growth rate per period / Starting value | 800 | 1,000 | 1,200 |
|---|---|---|---|
| 4.0% | 1,184.2 | 1,480.2 | 1,776.3 |
| 6.0% | 1,432.7 | 1,790.8 | 2,149 |
| 8.0% | 1,727.1 | 2,158.9 | 2,590.7 |
| 10.0% | 2,075 | 2,593.7 | 3,112.5 |
| 12.0% | 2,484.7 | 3,105.8 | 3,727 |
Exponential Growth: which assumptions move the result?
Exponential Growth: values and definitions
| Quantity and definition | Value used |
|---|---|
| Starting value | 1,000 |
| Growth rate per period — Use a negative rate for a decline. Doubling time applies only to a positive rate. | 8.0% |
| Number of periods | 10 |
| Ending value | 2,158.9 |
| Periods to double | 9 |
| Total growth | 115.9% |
Exponential Growth: half and double Growth rate per period
| Growth rate per period | Ending value |
|---|---|
| 4.0% | 1,480.2 |
| 8.0% | 2,158.9 |
| 16.0% | 4,411.4 |
Common error: Treating a decline as a growth rate in reverse. Falling 20 percent and then rising 20 percent does not get you back, because the second change applies to the smaller number.
Real quantities rarely grow at a constant rate indefinitely, since resources and demand limit growth, and early exponential phases often turn into S-curves. Small differences in the rate produce large differences over long periods. Negative rates model decay, and a rate of minus 100% or lower is not meaningful.
Discrete periods differ from continuous growth.
Algebra & Arithmetic calculators: Quadratic Formula Calculator, Fraction and Ratio Simplifier Calculator, Exponent Calculator, Root Calculator, Logarithm Calculator, Rounding Calculator.
Exponential growth adds a fixed percent of the current value each period. A value of 1,000 growing 8 percent a period is 2,158.92 after 10 periods, a rise of 115.9 percent. It doubles in about 9.01 periods, since the time to double depends on the rate and not on the starting value.
Use the same period for the rate and the count. An 8 percent yearly rate over 10 years is 10 periods, and an 8 percent monthly rate over 10 years is 120.
See the compound interest and future value calculators for money, the rule of 72 for doubling times, and the geometric sequence page for the same pattern as a list. The logarithm calculator solves for time, and the continuous compounding page covers the continuous limit.
What is exponential growth?
Growth in which the quantity increases by a constant percentage each period, so the increase itself gets larger over time.
How do I find the doubling time?
Divide the natural log of 2 by the natural log of one plus the growth rate. At 8% it is about 9 periods.
Can exponential growth continue forever?
No. Real systems run into limits, so growth usually slows, which is why many curves flatten into an S shape.