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How Exponential Growth Is Calculated: Formula, Rate, and Doubling Time

Understand the exponential growth formula and how to calculate population size, compound returns, or bacterial growth after any time period. Covers doubling time and half-life.

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The Exponential Growth Formula

N(t) = N₀ × e^(rt)

Where: - N(t) = quantity at time t - N₀ = initial quantity - e = Euler's number (≈ 2.71828) - r = growth rate (as a decimal, e.g., 0.05 for 5% per period) - t = time elapsed (in whatever unit r is per)

Alternative (percentage form): N(t) = N₀ × (1 + r)^t

The (1 + r)^t form is more intuitive for annual percentage growth; e^rt is the continuous compounding form. Both approach the same answer as compounding frequency increases.

Worked Examples

Example 1: Bacterial population Starting bacteria: 1,000 cells; doubling time: 20 minutes

r = ln(2) / 20 minutes = 0.693 / 20 = 0.03466 per minute

After 1 hour (60 minutes): N(60) = 1,000 × e^(0.03466 × 60) = 1,000 × e^2.08 = 1,000 × 8.0 = 8,000 cells

Example 2: Investment growth $10,000 invested at 7% annually for 20 years: N = 10,000 × (1.07)^20 = 10,000 × 3.870 = $38,700

Example 3: Population growth City population of 500,000 growing at 2.5% annually; population in 15 years: N = 500,000 × (1.025)^15 = 500,000 × 1.448 = 724,000

Doubling Time Formula

Doubling Time = ln(2) / r ≈ 0.693 / r

Or using the "Rule of 70": Doubling Time ≈ 70 / (rate in %)

Examples: | Growth Rate | Doubling Time | |------------|--------------| | 1% per year | 70 years | | 2% per year | 35 years | | 5% per year | 14 years | | 7% per year | 10 years | | 10% per year | 7 years | | 100% per period | 1 period |

Bacteria at 20-minute doubling: 70 / (100% per 20 min) = confirmed 20-minute doubling

Half-Life: Exponential Decay

Exponential decay uses the same formula but with negative rate (r < 0):

N(t) = N₀ × e^(−λt)

Half-life = ln(2) / λ ≈ 0.693 / λ

Example: Radioactive isotope with 5,730-year half-life (Carbon-14): λ = 0.693 / 5,730 = 0.0001209 per year

After 1,000 years: N = N₀ × e^(−0.0001209 × 1000) = N₀ × 0.8862 = 88.6% remains

Real-World Exponential Growth Applications

  • Compound interest: Money growing at fixed APY
  • Population dynamics: Human, animal, or bacterial populations
  • Epidemiology: R-number in disease spread (R > 1 = exponential growth)
  • Technology: Moore's law (transistor count doubling every ~2 years)
  • Nuclear decay: Radioactive material decay rates
  • Cooling/warming: Newton's law of cooling follows exponential approach to equilibrium

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Frequently Asked Questions

What is the exponential growth formula?
N(t) = N₀ × (1 + r)^t for discrete compounding, or N(t) = N₀ × e^(rt) for continuous growth. N₀ is starting value, r is growth rate per period, t is number of periods. For $10,000 at 6% for 5 years: 10,000 × (1.06)^5 = $13,382.
How do I calculate doubling time?
Doubling Time = ln(2) / r ≈ 0.693 / r. Or use Rule of 70: Doubling Time ≈ 70 / (rate as percent). At 7% growth: doubling time ≈ 70 / 7 = 10 years. At 2% growth: 35 years. For bacteria with 20-minute doubling time, they double every 20 minutes regardless of population size.
What is the difference between exponential and linear growth?
Linear growth adds a fixed amount each period (10, 20, 30, 40...). Exponential growth multiplies by a fixed factor each period (10, 20, 40, 80...). Initially similar, they diverge dramatically over time. A 7% annual return on $10,000 reaches $38,700 in 20 years (exponential) vs. $24,000 at $700/year flat (linear).
Does bacteria really grow exponentially?
Yes — under ideal conditions (unlimited nutrients, optimal temperature, no predation), bacteria double at a constant time interval (their doubling time). E. coli doubles every ~20 minutes in ideal conditions: starting from 1 cell, you'd theoretically have 4.7 trillion cells in 8 hours. Real conditions limit growth through resource depletion and waste accumulation.

Last updated 7/28/2026