1. Compound interest on $10,000 at 7% for 30 years
Compound interest follows FV = P × (1 + r)^t, where P is the principal, r the annual rate, and t the number of years. Plugging in P = $10,000, r = 7%, and t = 30 years gives FV = 10,000 × 1.07³⁰ = $76,122.55, a gain of $66,122.55 without adding another dollar. The balance grows to more than 7.6 times the original deposit purely from compounding.
2. Population of 5,000,000 growing 1.2% per year for 50 years
Population growth uses the same formula with r as the annual growth rate: P(t) = P₀ × (1 + r)^t. Starting at 5,000,000 people and growing 1.2% a year, 50 years brings the population to 5,000,000 × 1.012⁵⁰ = 9,078,114, an increase of about 81.6%. Small annual rates compound into large changes over half a century, so demographers watch percentage growth rates as closely as raw birth and death counts.
3. E. coli starting at 1,000 cells, 20-minute generation time, 6 hours
Bacterial growth by binary fission follows N(t) = N₀ × 2^(t / g), where g is the generation time. With a 20-minute generation time, 6 hours (360 minutes) equals 360 / 20 = 18 generations. Starting from 1,000 cells, the population reaches 1,000 × 2¹⁸ = 262,144,000 cells, a 262,144-fold increase in a single afternoon under ideal lab conditions with unlimited nutrients.
4. Caffeine 200 mg with 5-hour half-life, 12 hours later
Drug clearance follows the half-life formula N(t) = N₀ × (1/2)^(t / h), where h is the half-life. A 200 mg dose of caffeine with a 5-hour half-life leaves 200 × (1/2)^(12/5) = 200 × 0.1895 = 37.89 mg remaining after 12 hours. That is about 19% of the original dose, since 12 hours is a little under two and a half half-lives, 2.4 to be exact.
5. COVID-style epidemic with R₀ = 2.5 and 5-day generation, 30 days
Epidemic growth follows cases(t) = cases₀ × R₀^(t / g). Starting from 100 cases with R₀ = 2.5 and a 5-day generation time, 30 days is 30 / 5 = 6 generations, giving 100 × 2.5⁶ = 24,414 cases. The doubling time works out to g × ln(2) / ln(R₀) = 5 × ln(2) / ln(2.5) ≈ 3.78 days, meaning case counts roughly double every three and three-quarter days during this early exponential phase.
6. Saving $500/month at 6% for 30 years starting from $0
Regular savings use the future value of an annuity, FV = C × [((1 + i)^n − 1) / i], where C is the monthly deposit and i the monthly rate. Depositing $500 a month at 6% annual interest (i = 0.5% monthly) for 30 years (360 months) grows to $502,257.52. Of that total, $180,000 came directly from deposits and $322,257.52 came from compound interest, about 1.8 times as much as the money actually put in.
7. Carbon-14 sample with 25% remaining
Radiocarbon dating uses the decay formula solved for time: age = -h × log₂(fraction remaining), where h is the half-life of 5,730 years for carbon-14. A sample with 25% of its original carbon-14 remaining gives age = -5,730 × log₂(0.25) = -5,730 × (-2) = 11,460 years, exactly two half-lives, since one-quarter remaining means the sample has halved twice.
8. Logistic growth: P₀ = 50, K = 10,000, r = 50% per year, t = 25 years
Logistic growth follows P(t) = K / (1 + ((K − P₀) / P₀) × e^(-rt)), which adds a carrying capacity K to ordinary exponential growth. Starting from P₀ = 50 with K = 10,000 and r = 50% per year, 25 years gives P(25) = 10,000 / (1 + (9,950 / 50) × e^(-0.5 × 25)) = 9,993, about 99.9% of carrying capacity. Early growth looks exponential, but the curve flattens hard as the population approaches its ceiling.
FAQ
Are these examples real?
Yes, all eight examples use realistic numbers drawn from biology, finance, physics and epidemiology rather than invented figures. Each one is solved step by step using the standard exponential equation x(t) = x₀ × (1 + r)^t, or its continuous form x₀ × e^(kt) where the process compounds continuously. Working through all eight shows the same formula applying across very different real-world situations, from bacteria to bond interest.
Where should I start?
Start with example 1, compound interest, since it is the most familiar everyday use of exponential growth for most readers. From there, work through population growth, bacterial growth, drug decay, epidemic spread, savings, radiocarbon dating and logistic growth in any order that matches your interest. Each example stands on its own, so skipping ahead to the one closest to your own use case works just as well.
Can I change the numbers?
Yes, every example links directly to an interactive calculator pre-filled with that exact scenario's starting numbers. Moving any slider on the linked calculator instantly recalculates the result, the growth table and the chart, so you can test what-if questions like a higher interest rate or a shorter generation time. Nothing needs to be typed from scratch; the example simply becomes the calculator's starting point.
What level of math is required?
Basic arithmetic and comfort using a calculator are enough to follow every example on this page from start to finish. Logarithms only appear when solving for an unknown time period, such as finding when a population triples, and every calculator on this site computes those logarithms automatically. No calculus or algebra beyond rearranging one equation is needed to understand any of the eight worked problems.
How accurate are these projections?
The arithmetic in every example is exact, since each result comes directly from the stated formula with no rounding beyond display precision. Real-world accuracy instead depends on whether the underlying growth or decay rate actually stays constant over the full time period being modeled. Populations, investment returns and biological systems all drift over time, so long-range projections should be read as estimates, not guarantees.
Where can I learn more?
Each of the eight examples links to its own dedicated calculator page with a fuller explanation, additional FAQ questions and more worked numbers specific to that topic. The exponential growth formula reference page collects every discrete, continuous and decay variant of the formula in one place for quick comparison. Starting from either the examples or the formula reference eventually connects to every calculator on the site.