Example 1 — Continuous Growth
Initial value \( N_0 = 5{,}000 \), rate \( r = 0.06\,\mathrm{yr}^{-1} \), time \( t = 3\,\mathrm{yr} \).
Exponential Growth Calculator
Exponential Growth Calculator projects future values using rate, initial amount, continuous or discrete compounding, doubling time, and formula steps.
Starting amount at t = 0. Non-negative; supports math (e.g., 1e4).
Percent per period. Use negative for decay (e.g., −3).
Number of periods (years, months, etc.). Can be fractional (e.g., 2.5).
Enter any 3 fields and leave one blank; we solve the blank.
Equation Preview
Helping Notes
- Formula: \(x(t)=x_0(1+\tfrac{r}{100})^t\). r is percent per period (we divide your input by 100).
- To solve for r or t we use logarithms: \(r=100[(\tfrac{x(t)}{x_0})^{1/t}-1]\), \(t=\dfrac{\ln(x(t)/x_0)}{\ln(1+r/100)}\).
- Leave exactly one field blank (like Omni). All inputs accept math expressions (e.g., 1e3, 3/4).
Results
Solved value
All variables
Error
Steps
What Is an Exponential Growth Calculator?
An Exponential Growth Calculator models how a quantity increases when its growth rate is proportional to its current size. In continuous time, the differential equation
produces an exponential curve, where the growth rate \( r \) (per time unit) determines how fast the quantity grows. For discrete scenarios, such as monthly or yearly updates, the quantity increases by a fixed percentage per period. This calculator applies to populations, investments, biological cultures, and online viral content. It computes continuous and discrete growth, converts between parameters, and determines key metrics like doubling time or the time required to reach a target value.About the Exponential Growth Calculator
Enter an initial value \( N_0 \), growth rate (continuous \( r \) or discrete percentage \( g \)), and time horizon (\( t \) or number of periods \( n \)). The tool calculates projected quantities, doubling/halving times, and can solve for missing variables when a target value \( N \) is provided. Steps are displayed, including parameter conversion, formula substitution, and simplification. Units are flexible (people, dollars, cells), but time units must match the chosen rate. For decay, use negative \( r \) or \( g \).
How to Use the Exponential Growth Calculator
- Select a model: Continuous \( N(t) = N_0 e^{rt} \) or Discrete \( N_n = N_0 (1+g)^n \).
- Enter the initial value \( N_0 \), the growth rate \( r \) per time or \( g \) per period, and the time horizon.
- Optionally, provide a target value \( N \) to solve for unknown time or rate.
- Click calculate to view parameter conversions, substituted formulas, and final results.
- Compare growth scenarios using the doubling-time output and toggles for continuous vs discrete models.
Core Formulas
Continuous growth:
Discrete growth:
Rate conversions:
Doubling time and time to target:
Examples
Example 2 — Discrete Annual Percentage
Initial value \( N_0 = 12{,}000 \), growth \( g = 8\% \), periods \( n = 5 \) years.
Example 3 — Time to Reach a Target
Initial \( N_0 = 2\times10^6 \), target \( N = 1\times10^7 \), rate \( r = 0.12\,\mathrm{yr}^{-1} \).
Example 4 — Doubling Time
Growth rate \( r = 0.035\,\mathrm{yr}^{-1} \).
FAQs
What is the difference between continuous and discrete growth?
Continuous growth follows \( N(t) = N_0 e^{rt} \), while discrete growth uses \( N_n = N_0 (1+g)^n \). They are connected via \( r = \ln(1+g) \).
How do I compute doubling time?
For continuous growth: \( T_{\text{double}} = \ln 2 / r \). For discrete growth: \( n = \ln 2 / \ln(1+g) \).
Can this handle decay instead of growth?
Yes. Enter a negative \( r \) or \( g \) to model exponential decline.
Which rate should I enter: percent or per time?
Either works. The calculator converts between continuous \( r \) and discrete \( g \) automatically.
What units should I use?
Any units for \( N \) are valid. Ensure time units match the growth rate (e.g., years with \( r \) per year).
Why do continuous and discrete results differ?
Differences arise from compounding assumptions. They are small for low rates and short periods.
How do I solve for the rate from two observations?
Use \( r = (1/t) \ln(N/N_0) \) for continuous growth or \( g = (N/N_0)^{1/n} - 1 \) for discrete growth.