1,648.7213
648.7213
1.648721
x
13.8629
yr
-86.4086
%
1,648.7213
648.7213
1.648721
x
13.8629
yr
-86.4086
%
The Exponential Growth Calculator models continuous exponential growth using the formula $$P(t) = P_0 \cdot e^{rt}$$. This fundamental model describes processes where the growth rate is proportional to the current quantity — from population dynamics and compound interest to bacterial reproduction and viral spread.
Unlike linear growth (which adds a fixed amount each period), exponential growth accelerates over time because each increment is a percentage of an ever-larger quantity. This calculator computes final values, doubling times, and total growth for any initial value, rate, and time period.
The continuous exponential growth formula is:
$$P(t) = P_0 \cdot e^{rt}$$
where $$P_0$$ is the initial quantity, $$r$$ is the growth rate (as a decimal), $$t$$ is time, and $$e \approx 2.71828$$ is Euler's number.
Doubling time is the time required for the quantity to double:
$$t_d = \frac{\ln 2}{r} \approx \frac{0.693}{r}$$
The growth factor is the ratio $$\frac{P(t)}{P_0} = e^{rt}$$, and the total percent increase is $$(e^{rt} - 1) \times 100\%$$.
This model assumes continuous compounding. For discrete compounding (e.g., annual), use $$P(t) = P_0(1 + r)^t$$ instead.
The Final Value is the quantity after time $$t$$. The Growth Amount shows the absolute increase ($$P(t) - P_0$$). The Growth Factor tells you how many times larger the final value is compared to the initial. The Doubling Time is how long it takes for any quantity at this rate to double. The Total Percent Increase expresses the overall growth as a percentage.
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Results
A population of 1,000 growing at 5% continuously reaches ~1,649 after 10 years. Doubling time is ~13.86 years.
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Results
500 bacteria at 20%/hour continuous growth reach ~1,359 after 5 hours (a factor of e ≈ 2.718).
Linear growth adds a constant amount each period (e.g., +50/year). Exponential growth multiplies by a constant factor each period (e.g., ×1.05/year). Exponential growth starts slower but eventually surpasses any linear rate.
Continuous growth means the quantity increases at every instant, not just at discrete intervals. The rate $$r$$ in $$P_0 e^{rt}$$ is the instantaneous rate. A 5% continuous rate is slightly different from 5% annual compounding.
The Rule of 70 approximates doubling time as $$70 / r\%$$. This comes from $$\ln(2) \approx 0.693$$, so $$t_d = 0.693/r$$. Multiplying numerator and denominator by 100 gives $$69.3/r\% \approx 70/r\%$$.
Mathematically, yes — exponential growth has no upper bound. In reality, resources limit growth. The logistic model $$P(t) = K / (1 + Ae^{-rt})$$ adds a carrying capacity $$K$$ for more realistic modeling.
If the continuous rate is $$r_c$$, the equivalent discrete annual rate is $$r_d = e^{r_c} - 1$$. Conversely, $$r_c = \ln(1 + r_d)$$. For example, 5% continuous ≈ 5.127% discrete.
Population biology, compound interest, radioactive production, viral epidemics, technology adoption curves, inflation effects, and Moore's Law (transistor density doubling) all follow exponential growth patterns.
Roboculator Team
The Roboculator Team explains calculations, planning tools, and practical formulas in clear language for real-life situations.
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