Understanding growth rate and doubling time
Doubling time falls sharply as the growth rate rises, because the relationship is logarithmic rather than linear. The table below shows approximate doubling times for common growth rates, computed from t = ln(2) ÷ ln(1 + r).
| Growth rate per period | Exact doubling time (periods) | Rule-of-70 approximation |
|---|---|---|
| 1% | ≈ 69.66 | 70 |
| 2% | ≈ 35.00 | 35 |
| 5% | ≈ 14.21 | 14 |
| 10% | ≈ 7.27 | 7 |
| 20% | ≈ 3.80 | 3.5 |
- This calculator models discrete-period compounding (once per period), not continuous compounding. Continuous exponential growth uses P(t) = P₀ × eʳᵗ instead and produces slightly different results at the same nominal rate.
- A decay rate of 100% or more (factor ≤ 0) is not modeled — the value would reach zero or become undefined, so results are only returned for rates that keep the growth/decay factor strictly positive.
- The 'Rule of 70' (or 'Rule of 72' in finance) is a well-known mental-math approximation of the exact logarithmic doubling-time formula; it is most accurate for rates below roughly 10–15% per period.
What is exponential growth?
Exponential growth occurs when a quantity increases by a constant percentage rate at each equal time interval, so that the amount of increase itself grows larger over time — unlike linear growth, which adds the same fixed amount each period. Exponential decay is the mirror-image process: a quantity decreases by a constant percentage rate each period, with the amount of decrease shrinking over time as the base value shrinks.
The same mathematical formula, P(t) = P₀ × (1 ± r)ᵗ, describes both growth and decay: P₀ is the initial value, r is the rate per period expressed as a decimal, t is the number of periods, and the sign is positive for growth or negative for decay. This is the discrete-period compound growth formula, mathematically identical to compound interest with no additional contributions.
Exponential models appear throughout the sciences: population growth under unlimited resources, radioactive decay, compound interest, the spread of infectious disease in its early phase, and Moore's law for transistor density all follow this same exponential form, though the underlying rate r and its real-world drivers differ across each domain.
How to use this exponential growth calculator
- Enter the initial value — the starting amount of the quantity you are modeling (e.g. population, an investment balance, or a substance quantity).
- Enter the growth or decay rate as a percentage per period (e.g. 5 for a 5% rate).
- Enter the number of periods to project forward. Periods must use the same time unit as the rate (e.g. an annual rate needs a number of years).
- Select 'Growth' if the value increases each period, or 'Decay' if it decreases. Read the final value, the total change from the initial value, and the doubling or halving time.
The exponential growth and decay formula
The core formula is P(t) = P₀ × (1 + r)ᵗ for growth, or P(t) = P₀ × (1 − r)ᵗ for decay, where P₀ is the initial value, r is the rate per period as a decimal, and t is the number of periods elapsed. Worked example (growth): with P₀ = 1000, r = 5% = 0.05, t = 10 periods: P(10) = 1000 × 1.05¹⁰ ≈ 1000 × 1.628895 ≈ 1628.89. The total change is 1628.89 − 1000 = 628.89.
Doubling time (growth) and halving time (decay) answer 'how many periods until the value doubles/halves?' and are derived by solving (1 ± r)ᵗ = 2 (or 0.5) for t using logarithms: t = ln(2) ÷ |ln(1 ± r)|. For the 5% growth example, doubling time = ln(2) ÷ ln(1.05) ≈ 0.693147 ÷ 0.048790 ≈ 14.21 periods. This is the exact form of the commonly cited 'Rule of 70' approximation (70 ÷ rate% ≈ 14 periods for a 5% rate), which is a rounded shortcut for the same logarithmic formula.
For decay, the same logarithmic approach gives the halving time. Example: a 3% decay rate (factor 0.97) has a halving time of ln(0.5) ÷ ln(0.97) ≈ 22.76 periods — the number of periods for the quantity to fall to half its starting value.
Common mistakes
- Mismatching the rate's time unit and the number of periods — an annual growth rate must be paired with a number of years, not months or days, unless the rate is first converted.
- Treating exponential growth as if it were linear — doubling the number of periods does not double the final value; it squares the growth factor's contribution, producing much larger changes than linear intuition suggests.
- Entering a decay rate above 100% (e.g. 150%), which produces a negative or undefined growth factor and is not a valid model of proportional decay.
- Confusing 'doubling time' with 'the time to reach a specific target value' — doubling time is a fixed property of the rate alone and does not depend on the starting value.
- Using the Rule of 70/72 approximation and expecting it to exactly match the logarithmic calculation — the rule is a close approximation, not an exact formula, and diverges more at higher rates.
Perguntas frequentes
What is the formula for exponential growth?
The exponential growth formula is P(t) = P₀ × (1 + r)ᵗ, where P₀ is the initial value, r is the growth rate per period expressed as a decimal, and t is the number of periods. For example, $1000 growing at 5% per period for 10 periods gives 1000 × 1.05¹⁰ ≈ $1628.89.
How do you calculate doubling time?
Doubling time is calculated as t = ln(2) ÷ ln(1 + r), where r is the growth rate as a decimal. For a 5% growth rate, doubling time = ln(2) ÷ ln(1.05) ≈ 14.21 periods. This tells you how many periods it takes for a quantity growing at that rate to double in size, regardless of its starting value.
What is the difference between exponential growth and linear growth?
Linear growth adds the same fixed amount each period (e.g. +10 units every period), while exponential growth adds a fixed percentage of the current value each period, so the absolute increase grows larger over time. A quantity growing exponentially eventually outpaces any quantity growing linearly, no matter how large the linear increment.
What is the Rule of 70?
The Rule of 70 is a mental-math shortcut estimating doubling time as 70 divided by the growth rate percentage. For a 5% growth rate: 70 ÷ 5 = 14 periods, close to the exact logarithmic result of 14.21 periods. It is an approximation of the exact formula t = ln(2) ÷ ln(1 + r) and is most accurate at lower growth rates.
How is exponential decay different from exponential growth?
Exponential decay uses a rate applied as a reduction rather than an increase: P(t) = P₀ × (1 − r)ᵗ. Both growth and decay share the same mathematical structure — a fixed percentage change compounding each period — but decay approaches zero over time while growth increases without bound.
Can the growth rate be negative in the 'growth' mode?
Yes — entering a negative rate in growth mode produces a shrinking value, which is mathematically equivalent to decay. The 'growth' and 'decay' modes exist mainly for clarity of interpretation and for computing the correct doubling-time versus halving-time label.
Referências
- NIST Digital Library of Mathematical Functions (DLMF), §4.2 Elementary Properties of the Exponential Function. dlmf.nist.gov.
- Boyce WE, DiPrima RC. Elementary Differential Equations and Boundary Value Problems. 11th ed. Wiley, 2017. (Exponential growth/decay models.)
- Weisstein EW. 'Doubling Time' and 'Exponential Growth.' MathWorld — A Wolfram Web Resource. mathworld.wolfram.com.