Exponential Growth Calculator
Calculate growth amounts, doubling time, and growth rates for population, investment, and other exponential growth processes
N(t) = N₀ × ekt
Exponential growth formula where k is the growth rate and t is time
Starting quantity (must be positive)
Rate of growth (per unit time)
Time elapsed (same unit as k)
Time for quantity to double
Quick Examples
Invalid Input
Final Amount After time units
growth (× original)
Doubling Time for k =
time units for quantity to double
Growth Rate for t₂ =
per time unit
Growth Visualization
Initial Amount
Final Amount
Doubling Time
# of Doublings
Step-by-Step Solution
Identify values
N₀ = , k = , t =
Apply the formula
N(t) = N₀ × ekt = × e( × )
Calculate exponent
kt = × =
Final result
N() =
Common Growth Rates Reference
| Scenario | Growth Rate | Doubling Time |
|---|---|---|
| Bacterial (optimal) | ~69% per hour | ~1 hour |
| Investment (7%) | 7% per year | ~10 years |
| World Population | ~1.1% per year | ~63 years |
| Moore's Law | ~41% per year | ~18 months |
| Inflation (3%) | 3% per year | ~23 years |
| Rule of 72 (Quick Est.) | x% per period | ~72/x periods |
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About Exponential Growth Calculator
What is Exponential Growth?
Exponential growth describes a process where a quantity increases over time at a rate proportional to its current value. This mathematical model is fundamental in biology, economics, finance, and many other fields.
Mathematical Formula
The exponential growth formula is:
N(t) = N₀ × e^(kt)
Where:
- N(t) = Quantity after time t
- N₀ = Initial quantity (at time t = 0)
- e = Euler's number (approximately 2.71828)
- k = Growth rate constant (positive value)
- t = Time elapsed
Doubling Time Relationship
The doubling time (t₂) is the time required for a quantity to double in value:
t₂ = ln(2) / k ≈ 0.693 / k
Converting between doubling time and growth rate:
- k = ln(2) / t₂
- t₂ = ln(2) / k
Alternative Forms
Using Percentage Growth Rate
N(t) = N₀ × (1 + r)^t
Where r is the growth rate per time period (e.g., 0.05 for 5%).
Using Doubling Time
N(t) = N₀ × 2^(t/t₂)
Applications
1. Population Growth
Bacterial populations, human populations in ideal conditions, and invasive species often follow exponential growth initially.
2. Compound Interest
Money invested with compound interest grows exponentially: A = P × e^(rt) (continuous compounding)
3. Viral Spread
In the early stages, disease spread often follows exponential growth patterns.
4. Cell Division
Cell populations doubling at regular intervals exhibit exponential growth.
5. Technology Adoption
New technology adoption curves often show initial exponential growth.
Common Growth Rates
| Scenario | Growth Rate | Doubling Time |
|---|---|---|
| Bacterial division (optimal) | 100% per hour | ~42 minutes |
| Human population (2%) | 2% per year | ~35 years |
| Investment (7% annual) | 7% per year | ~10 years |
| Moore's Law (transistors) | 41% per year | ~18 months |
| Inflation (3%) | 3% per year | ~23 years |
Key Properties
- J-Curve Shape: Growth starts slowly but accelerates dramatically
- Unlimited Growth: In theory, continues forever (limited by resources in reality)
- Constant Percentage: Same percentage increase in each equal time period
- Rapid Acceleration: Quantity can become very large very quickly
Important Notes
- Real-world growth is often limited by resources (logistic growth)
- The growth rate k must be positive for growth (negative k means decay)
- Time and growth rate must use consistent units
- Small differences in growth rates lead to large differences over time