Many things grow by a roughly constant percentage rather than a constant amount: app users, social media followers, bacteria in a culture, an investment, the spread of an early epidemic. When growth is exponential, the ratio between the values at the start and end of a period tells you everything – a channel that went from 5,000 to 8,000 subscribers in 6 months grew by a factor of 1.6, or about 8.15% a month.
Once you know the rate per period you can project forward: at 8.15% a month, 8,000 subscribers become 8,000 × 1.6² = 20,480 in another 12 months, and reaching 50,000 takes about 23.4 months. The calculator also gives the doubling time and converts the rate to every common period, so you can compare a daily growth rate with an annual one.
Treat the result as a projection, not a promise. Real growth usually slows as markets saturate or resources run out (logistic growth), so exponential forecasts are most reliable over short horizons and become optimistic the further you extend them.
How to use the Exponential Growth Prediction Calculator
- Past: starting value
- Past: ending value
- Past: time elapsed
- Time unit
- Predict
- Future: starting value — Usually the latest value you have.
- Future: time ahead (same unit)
- Future: target value
Results update as you type, and everything is calculated in your browser.
How it works
The past growth factor F/I is spread evenly over the elapsed time to get the rate per period. Rates for other periods use 1 day, 7 days, 30.4375 days per month and 365.25 days per year. The future value multiplies the new starting value by the growth factor raised to the number of periods, and the time to a target inverts that formula with logarithms.
Worked example
A channel grew from 5,000 to 8,000 subscribers in 6 months. The growth factor is 1.6, so the monthly rate is 1.6^(1/6) − 1 ≈ 8.148%, the daily rate 1.6^(1/182.6) − 1 ≈ 0.258% and the annual rate 1.6² − 1 = 156%. Starting from today’s 8,000, twelve more months give 8,000 × 1.6^(12/6) = 20,480 subscribers. Reaching 50,000 takes 6 × ln(6.25) ÷ ln(1.6) ≈ 23.4 months, and the doubling time is 6 × ln 2 ÷ ln 1.6 ≈ 8.85 months.
Doubling times for common growth rates
| Rate per period | Doubling time (periods) | Rule of 72 estimate |
|---|---|---|
| 1% | 69.66 | 72.0 |
| 2% | 35.00 | 36.0 |
| 5% | 14.21 | 14.4 |
| 7% | 10.24 | 10.3 |
| 10% | 7.27 | 7.2 |
| 15% | 4.96 | 4.8 |
| 20% | 3.80 | 3.6 |
| 50% | 1.71 | 1.4 |
Inputs explained
| Field | What to enter |
|---|---|
| Past: starting value | A number |
| Past: ending value | A number |
| Past: time elapsed | A number |
| Time unit | Choose one: Days, Weeks, Months, Years |
| Predict | Choose one: Value after a given time, Time to reach a target value |
| Future: starting value | A number. Usually the latest value you have. |
| Future: time ahead (same unit) | A number |
| Future: target value | A number |
What each result means
- Predicted value = [Future: starting value]×([Past: ending value]÷[Past: starting value])^([Future: time ahead (same unit)]÷[Past: time elapsed])
- Time to reach the target (in the chosen unit) = [Past: time elapsed]×ln([Future: target value]÷[Future: starting value])÷ln([Past: ending value]÷[Past: starting value])
- Growth rate per time unit = (([Past: ending value]÷[Past: starting value])^(1÷[Past: time elapsed]) – 1)×100
- Equivalent daily rate = (([Past: ending value]÷[Past: starting value])^(1÷([Past: time elapsed]×[Time unit])) – 1)×100
- Equivalent weekly rate = (([Past: ending value]÷[Past: starting value])^(7÷([Past: time elapsed]×[Time unit])) – 1)×100
- Equivalent monthly rate = (([Past: ending value]÷[Past: starting value])^(30.4375÷([Past: time elapsed]×[Time unit])) – 1)×100
- Equivalent annual rate = (([Past: ending value]÷[Past: starting value])^(365.25÷([Past: time elapsed]×[Time unit])) – 1)×100
- Continuous growth rate per time unit = ln([Past: ending value]÷[Past: starting value])÷[Past: time elapsed]×100
- Doubling time (halving time if negative) = [Past: time elapsed]×ln(2)÷ln([Past: ending value]÷[Past: starting value])
- Past growth factor (end ÷ start) = [Past: ending value]÷[Past: starting value]
Formula
Growth factor per period: 1 + r = (F / I)¹/t) · Future value = V₀ × (1 + r)^t₂ · Time to target = ln(target / V₀) ÷ ln(1 + r) · Doubling time = ln 2 ÷ ln(1 + r)
Worked example
Inputs — Past: starting value: 5,000; Past: ending value: 8,000; Past: time elapsed: 6; Time unit: Months; Predict: Value after a given time; Future: starting value: 8,000; Future: time ahead (same unit): 12.
Results — Predicted value = 20,480; Growth rate per time unit = 8.1484 %; Equivalent daily rate = 0.2577 %; Equivalent weekly rate = 1.8178 %; Equivalent monthly rate = 8.1484 %; Equivalent annual rate = 156 %; Continuous growth rate per time unit = 7.8334 %; Doubling time (halving time if negative) = 8.849; Past growth factor (end ÷ start) = 1.6.
Key terms
- Growth factor
- The ratio of the ending to the starting value over a period.
- Growth rate
- The percentage increase per time unit: growth factor per unit minus 1.
- Doubling time
- The time needed for the value to double at a constant growth rate.
- Continuous growth
- Growth compounded at every instant, described by V = V₀·e^(kt).
Frequently asked questions
How do I calculate an exponential growth rate?
r = (final ÷ initial)^(1/t) − 1. From 5,000 to 8,000 in 6 months: 1.6^(1/6) − 1 ≈ 8.15% per month.
How do I convert a monthly growth rate to an annual one?
Compound it: (1 + monthly)¹² − 1. 8.15% a month is 1.0815¹² − 1 ≈ 156% a year – not 12 × 8.15% = 97.8%.
How long until a value doubles?
Doubling time = ln 2 ÷ ln(1 + r). At 8.15% a month it is about 8.85 months; the rule of 72 gives a quick estimate (72 ÷ 8.15 ≈ 8.8).
Does it work for decline?
Yes. If the ending value is smaller the rate is negative and the "doubling time" becomes a negative halving time.
What is the difference between the periodic and the continuous rate?
The periodic rate compounds once per unit; the continuous rate k = ln(F/I) ÷ t compounds constantly. Both give the same final value.
How reliable is an exponential forecast?
Good for short horizons when conditions stay the same. Most real growth slows down eventually, so long forecasts usually overshoot.
Why is the time to target blank?
The target must lie in the direction of growth: you cannot reach a higher target with a shrinking value, or a lower one with a growing value.