Exponential Growth Calculator

Enter a starting value, growth rate, and time elapsed to find the final value, total growth, percent increase, and doubling time using y = y₀(1+r)^t or y = y₀e^(rt).

Quick Facts

Periodic growth
y = y₀(1 + r)ᵗ
Compounds a fixed number of times per period, like annual compound interest.
Continuous growth
y = y₀e^(rt)
Uses Euler's number e; the limit of periodic compounding as intervals grow infinite.
Doubling time
t₂ = ln(2) / ln(1+r) or ln(2) / r
Rule of 70 approximation: t₂ ≈ 70 / (r as a percent).

Your Results

Calculated
Final Value
-
y(t) from the selected growth model
Total Growth (Δy)
-
y(t) − y₀
Percent Increase
-
(y(t) / y₀ − 1) × 100%
Doubling Time
-
Time for the value to double at this rate

Ready

Enter an initial value, growth rate, and time elapsed, then press Calculate.

Formula and Method for Exponential Growth

Exponential growth describes a quantity that increases at a rate proportional to its current size, so the bigger it gets, the faster it grows. There are two standard ways to model it. Periodic (discrete) growth compounds a fixed number of times per period: y = y₀(1 + r)ᵗ, where y₀ is the initial value, r is the growth rate per period as a decimal, and t is the number of periods elapsed. Continuous growth compounds at every instant using Euler's number e: y = y₀e^(rt). This calculator also reports the total growth, percent increase, and doubling time for either model.

How the calculation works

Enter the initial value y₀, the growth rate r as a percentage per period, the time elapsed t, and pick a growth model. The rate is converted to a decimal (r ÷ 100) before use. For the periodic model, the calculator raises (1 + r) to the power t and multiplies by y₀; for the continuous model, it multiplies y₀ by e raised to the power (r × t). Total growth is simply the final value minus the initial value, and percent increase is that difference divided by the initial value, times 100. Doubling time — how long it takes the quantity to double at the current rate — is ln(2) / ln(1 + r) for periodic growth or ln(2) / r for continuous growth; it is only defined when the growth rate is positive.

Common mistakes

  • Entering the rate as a decimal instead of a percent: this calculator expects r as a percentage (e.g., 5 for 5%), not 0.05 — entering 0.05 would model a 0.05% rate, not 5%.
  • Mixing up periodic and continuous models: compound interest paid annually or monthly is periodic; population or radioactive processes are often modeled as continuous. The two give slightly different results for the same nominal rate.
  • Forgetting that negative rates model decay: a growth rate below 0% shrinks the quantity toward zero instead of growing it — useful for depreciation or decay, but easy to enter by accident.
  • Rounding the rate too aggressively: small rounding errors in r compound over many periods and can meaningfully shift long-horizon projections.

Real-world applications

  • Compound interest and investment growth projections, where r is the periodic interest rate.
  • Population growth modeling over generations or years.
  • Bacterial or viral spread during the early, unconstrained phase of growth.
  • Estimating how long an investment, population, or metric takes to double using the doubling-time formula.

Frequently Asked Questions

What is the formula for exponential growth?
The periodic (discrete) exponential growth formula is y = y₀(1 + r)ᵗ, where y₀ is the initial value, r is the growth rate per period expressed as a decimal, and t is the number of periods elapsed. The continuous-growth version is y = y₀e^(rt), which uses Euler's number e and applies when growth compounds continuously rather than at fixed intervals.
What is the difference between periodic and continuous exponential growth?
Periodic growth compounds a fixed number of times per period, such as annual compound interest, using y = y₀(1+r)ᵗ. Continuous growth compounds infinitely often within each instant, using y = y₀e^(rt), which is the limit of periodic compounding as the number of compounding intervals approaches infinity. For the same nominal rate, continuous growth produces a slightly larger final value than periodic growth.
How do I calculate doubling time?
For periodic growth, doubling time is t = ln(2) / ln(1+r). For continuous growth, it simplifies to t = ln(2) / r. A quick estimate is the Rule of 70: divide 70 by the growth rate as a percentage (70 / r%) to approximate the number of periods needed to double.
Can exponential growth model a decline instead of growth?
Yes. Using a negative growth rate (r < 0) in the same formulas models exponential decay instead of growth, such as radioactive decay or depreciation. The formulas are identical; only the sign and magnitude of r change, and the value shrinks toward zero over time rather than increasing.

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