Exponential Growth

The defining equation

Exponential growth is the unique solution to the differential equation:

dy/dt = k · y

"The rate of change is proportional to the current value." Solving this equation gives:

y(t) = y₀ · e^(kt)

where y₀ is the initial value, k is the growth rate (positive for growth, negative for decay), and t is time.

The discrete analog (growth in steps, not continuously) is:

y(t) = y₀ · (1 + r)^t

where r is the per-period growth rate. As periods become infinitely small, the discrete formula approaches the continuous one. They're equivalent for sufficiently fine time-stepping; (1 + r)^t ≈ e^(rt) when r is small.

The doubling-time property

For exponential growth, the time to double is constant — independent of starting value. From y(t+T)/y(t) = 2 and y(t) = y₀·e^(kt):

e^(k(t+T)) / e^(kt) = e^(kT) = 2
kT = ln(2)
T = ln(2) / k ≈ 0.693 / k

So at growth rate k = 0.07 per year (continuous), doubling time is 0.693 / 0.07 ≈ 9.9 years. The famous "rule of 72" approximates this — 72/7 ≈ 10.3 years — using a slightly larger numerator that's easier to divide mentally.

Growth rate	Rule of 72 doubling	Exact doubling
1% / year	72 years	69.7 years
3%	24 years	23.4 years
5%	14.4 years	14.2 years
7%	10.3 years	10.2 years
10%	7.2 years	7.3 years
20%	3.6 years	3.8 years

Worked examples

Example 1 — bacterial growth

A culture starts with 100 bacteria, doubling every 30 minutes. How many after 4 hours (240 minutes)?

Number of doublings = 240 / 30 = 8
y(240) = 100 · 2⁸ = 100 · 256 = 25,600 bacteria

Or in continuous form — k = ln(2)/30 ≈ 0.0231 per minute. y(240) = 100·e^(0.0231·240) = 100·e^5.55 ≈ 25,600. Same answer.

Example 2 — compound interest

$1,000 invested at 5% annual interest, compounded continuously, for 20 years.

y(20) = 1000 · e^(0.05 · 20) = 1000 · e¹ = 1000 · 2.718 = $2,718

The same investment compounded annually — y(20) = 1000 · 1.05²⁰ = 1000 · 2.653 = $2,653. Continuous compounding gives more — the difference is the value of "instant reinvestment."

Example 3 — radioactive decay

Carbon-14 has a half-life of 5,730 years. If a sample originally had 1.0 g of C-14 and now has 0.25 g, how old is it?

0.25 = 1.0 · (1/2)^(t/5730)
log(0.25) = (t/5730) · log(0.5)
log(1/4) = -2 · log(2) = -2 · log(2)
log(1/2) = -log(2)
So: -2·log(2) = (t/5730) · -log(2)
t/5730 = 2
t = 11,460 years

Two half-lives — sample is 11,460 years old. Carbon dating uses this principle on archaeological remains.

Exponential vs other growth types

Type	Formula	10× input gives	Examples
Constant	y = c	same output	Independent of input
Logarithmic	y = log(x)	+1 output	Algorithm depth, sensory perception
Linear	y = ax	10× output	Rate-of-pay × hours
Polynomial	y = x^n	10ⁿ× output	Area, volume, surface scaling
Exponential	y = a · b^x	b¹⁰× output	Population, compound interest, viral spread
Factorial	y = x!	vastly more	Permutations, brute-force combinations

Exponential growth eventually dominates any polynomial. For large enough x, 2^x exceeds x¹⁰⁰. But "large enough" can be enormous — 2^100 only exceeds 100^100 at x ≈ 1000+. The asymptotic dominance is real; the crossover point depends on constants.

When exponential growth stops — the logistic curve

Pure exponential growth requires unlimited resources. Real systems have constraints (finite environment, susceptible population, market size). The logistic equation models this:

dy/dt = k·y·(1 − y/K)

where K is the carrying capacity. When y is small (y << K), the (1 − y/K) factor is ≈ 1 and growth is exponential. When y approaches K, the factor approaches 0 and growth slows. Solution is the S-curve:

y(t) = K / (1 + ((K − y₀)/y₀) · e^(−kt))

Famous applications — bacterial growth (food limit), pandemic spread (susceptible population), technology adoption (market saturation), Moore's law slowdown.

JavaScript: simulating exponential growth and decay

// Continuous exponential
function continuousExp(y0, k, t) {
  return y0 * Math.exp(k * t);
}

// Discrete exponential (annual compound)
function discreteExp(y0, r, t) {
  return y0 * Math.pow(1 + r, t);
}

// Doubling time
function doublingTime(k) {
  return Math.LN2 / k;  // Math.LN2 = 0.693...
}

// Half-life
function halfLife(k) {
  return Math.LN2 / Math.abs(k);
}

// Rule of 72 — quick estimate
function ruleOf72(percentRate) {
  return 72 / percentRate;
}

// Logistic curve
function logistic(t, K, y0, k) {
  return K / (1 + ((K - y0) / y0) * Math.exp(-k * t));
}

// $1000 at 7% for 30 years, continuous
continuousExp(1000, 0.07, 30);  // 8166.17

// 1g of Carbon-14 after 20,000 years (half-life 5730)
const k_c14 = Math.LN2 / 5730;  // 0.000121
continuousExp(1, -k_c14, 20000);  // 0.0875g remaining

Where exponential growth shows up

• Population biology. Bacterial colonies, disease outbreaks, invasive species — all start exponentially. Logistic carrying capacity eventually constrains.
• Compound interest and finance. Money invested at constant rate grows exponentially. Inflation erodes purchasing power exponentially. Mortgages amortize in patterns derived from the exponential-decay equation.
• Radioactive decay and pharmacokinetics. Half-life is the universal description of first-order decay. Drug clearance, isotope decay, capacitor discharge — all follow the same math with negative growth rate.
• Moore's Law and technology. Transistor density doubles every ~2 years. Storage capacity, network bandwidth, and computational throughput follow similar exponentials, though all eventually slow due to physical limits.
• Viral content / network effects. Each share generates more shares; growth is exponential in the early phase. Saturates when the audience runs out.
• Information theory. Number of distinct messages of length n is exponential in n. Cryptographic key strength is measured in bits — each bit doubles the search space.

Common mistakes

• Confusing exponential with quadratic. 2^x grows much faster than x². They look similar at first; for large x, exponential dominates by orders of magnitude.
• Using linear extrapolation on exponential trends. "It went up 10% last month, so it'll go up 10 by next month" — wrong; it'll go up 10% of the new total. Compounds matter.
• Forgetting that real systems become logistic. Pure exponential growth doesn't last. Whether the limiting factor is resources, market size, or fundamental physics, real systems hit ceilings.
• Mixing discrete and continuous formulas. y₀·(1+r)^t and y₀·e^(rt) give different numerical answers; only the latter is the "instantaneous compounding" model. Don't substitute one for the other without knowing why.
• Negative bases. a^x for negative a is only well-defined for integer x (and even then, alternating signs). For continuous models, write decay as e^(-kt) with positive k, not (-1)^t or similar.
• Rule of 72 for very high or low rates. The approximation breaks down outside ~5-15% range. For rate-of-72-times-period to give doubling time, the rate should be moderate. 100% rate doubles in 1 period (72/100 = 0.72 — wrong); 0.1% takes 720 periods (close to actual 693). Use exact ln(2)/k for high-precision work.