Exponential Growth Calculator
Calculate exponential growth and decay with step-by-step solutions
Exponential Growth Calculator
Result
Step-by-Step Solution
Formula Types
Percentage Form
x(t) = x₀ × (1 + r/100)^t
Most common, r in percentage
Exponential Form
x(t) = x₀ × e^(kt)
Natural exponential, k is decay constant
Relationship
r = 100 × (e^k - 1)
k = ln(1 + r/100)
Key Properties
Growth vs Decay
r > 0: growth, r < 0: decay
Doubling Time
t = ln(2) / ln(1 + r/100)
Half-Life
t = ln(0.5) / k (for decay)
Continuous Growth
Exponential form models continuous processes
Understanding Exponential Growth and Decay
What is Exponential Growth?
Exponential growth occurs when a quantity increases by a fixed percentage in each time period. The distinguishing feature is that the rate of change is proportional to the current amount.
Key Characteristics
- •Constant percentage rate of change
- •J-shaped or reverse J-shaped curve
- •Rapid increase (or decrease) over time
- •Multiplicative process
Formula Comparison
Percentage Form
x(t) = x₀ × (1 + r/100)^t
Used for discrete compounding periods
Example: Annual population growth
Exponential Form
x(t) = x₀ × e^(kt)
Used for continuous processes
Example: Radioactive decay, bacterial growth
Real-World Applications
Population Growth
Bacterial cultures, human populations, animal populations
Formula: P(t) = P₀ × (1 + r)^t
Financial Growth
Compound interest, investment returns, inflation
Formula: A(t) = P × (1 + r/n)^(nt)
Radioactive Decay
Nuclear decay, drug metabolism, carbon dating
Formula: N(t) = N₀ × e^(-λt)
Important Notes
- • Exponential growth is often unsustainable in real-world scenarios due to resource limitations
- • For decay, ensure the growth rate is between -100% and 0%
- • The exponential model assumes a constant rate of change
- • Time can be negative to model past values