This page collects every variant of the exponential growth and decay formula in one reference. Each form links to a dedicated calculator with its own worked examples, tables and FAQ; use this page to see how the variants relate to each other and convert between them, not as a substitute for the deeper walkthroughs on those pages.
1. Discrete exponential growth
x(t) = x₀ × (1 + r)t
Example: $1,000 at 5% per year for 10 years → 1,000 × 1.05¹⁰ = $1,628.89. This is the everyday compounding formula behind savings accounts and annual raises; the exponential growth calculator extends it with a chart and step-by-step breakdown.
2. Continuous exponential growth
x(t) = x₀ × ekt
Example: $1,000 at k = 5% continuous for 10 years → 1,000 × e^0.5 = $1,648.72. Finance and biology both lean on this form when growth is effectively continuous rather than once per period; see the continuous growth calculator for the discrete-vs-continuous comparison.
3. Discrete exponential decay
x(t) = x₀ × (1 − r)t
Example: 100 mg drug with 10% per-hour decay for 5 hours → 100 × 0.9⁵ = 59.05 mg. Depreciation schedules and periodic drug elimination use this form; the exponential decay calculator works through remaining-amount tables.
4. Continuous exponential decay
x(t) = x₀ × e−kt
Example: 100 mg with k = 0.10 per hour for 5 hours → 100 × e^(−0.5) = 60.65 mg. Radioactive decay and pharmacokinetics prefer this continuous form; the decay rate calculator solves for k from two measurements.
5. Doubling time
t₂ = ln(2) / ln(1 + r) (discrete) or ln(2) / k (continuous). The doubling time calculator compares this exact value against the Rule of 70 mental-math shortcut across a full table of rates.
6. Half-life
t½ = ln(2) / k. The half-life calculator applies this to radiocarbon dating and drug dosing with worked examples.
7. Growth rate from two points
r = (x(t) / x₀)1/t − 1. The growth rate calculator is built around this formula, also known as CAGR when t is measured in years.
8. Logistic growth
P(t) = K / (1 + ((K − P₀) / P₀) × e−rt). The logistic growth calculator charts how this S-curve approaches the carrying capacity K over time.
Variable definitions
The table below defines every symbol used across the eight formulas above, since different fields, finance, biology and physics, traditionally use different letters for the same role.
| Symbol | Meaning |
|---|---|
| x₀, P₀, N₀ | Initial value |
| x(t), P(t), N(t) | Value at time t |
| r | Periodic rate (decimal or %) |
| k | Continuous rate constant |
| t | Time elapsed |
| K | Carrying capacity (logistic) |
| e | Euler's number ≈ 2.71828 |
| t½ | Half-life |
| t₂ | Doubling time |
Comparison of forms
| Form | Equation | When to use |
|---|---|---|
| Discrete growth | x₀ (1 + r)^t | Annual rates, compound interest |
| Continuous growth | x₀ e^(kt) | Calculus, bond yields, biology |
| Discrete decay | x₀ (1 − r)^t | Periodic loss, depreciation |
| Continuous decay | x₀ e^(-kt) | Radioactive, drug elimination |
| Logistic | K / (1 + ((K−P₀)/P₀)e^(-rt)) | Bounded systems |
Common formula mistakes
The most frequent error across all eight forms is forgetting to convert a percentage into a decimal before substituting it: plugging 5 into (1 + r)^t instead of 0.05 turns modest growth into an absurd 6^t explosion. The second most common mistake is mixing r and k without converting between them, since a rate copied from an annual compounding table straight into a continuous-growth formula silently understates or overstates the result by the gap between ln(1 + r) and r itself. A third, subtler error is applying a per-year rate to a number of months without first converting the rate or the time unit to match, which is why every calculator on this site labels its time input with the same unit as its rate input.
How to choose between discrete and continuous
When a source explicitly states how often compounding happens, annually, monthly, daily, use the matching discrete form with r as that period's rate. When a process is described as changing constantly rather than at fixed intervals, population growth, radioactive decay, drug concentration in the bloodstream, the continuous form with rate constant k is the more natural fit and simplifies any calculus performed on it afterward. In practice the two forms rarely disagree by more than a percentage point or two at typical real-world rates, so choosing the form that matches how the underlying process is usually described, rather than forcing a conversion, keeps the model easiest to communicate to someone else.
FAQ
Which formula should I use?
Use the discrete form x₀(1 + r)^t when growth is measured once per period, such as annual salary raises or yearly compound interest. Use the continuous form x₀e^(kt) when the process updates continuously, such as population growth modeled with calculus or continuously compounded bonds. Both produce nearly identical results for small rates; at r = 5%, discrete gives 1.6289 after 10 periods versus 1.6487 continuous, a difference of about 1.2%.
Are r and k the same?
No, they measure the same underlying growth but on different time bases. The continuous rate k relates to the discrete rate r by k = ln(1 + r), so a 5% discrete annual rate equals k ≈ 0.04879 continuous. Converting the other direction uses r = e^k − 1. Mixing the two without converting is a common source of small errors in financial models.
How do I convert percent to decimal?
Divide the percent value by 100 to get the decimal form used in every formula on this page. 5% becomes 0.05, 12% becomes 0.12, and -3% becomes -0.03. The decimal form is what actually gets substituted for r or k; plugging the raw percent number into (1 + r)^t instead of (1 + 0.05)^t is one of the most common calculator mistakes.
What is the doubling time formula?
Doubling time is t₂ = ln(2) / ln(1 + r) for the discrete form, or ln(2) / k for the continuous form, both derived by setting x(t) / x₀ = 2 and solving for t. At r = 7%, t₂ ≈ 10.24 periods. The doubling time calculator works through the Rule of 70 shortcut and more real-world examples in depth.
What is the half-life formula?
Half-life is t½ = ln(2) / k for continuous decay, or t½ = ln(2) / ln(1 / (1 − r)) for a discrete decay rate r, both found by setting the remaining fraction to one half and solving for t. A process with k = 0.1 per hour has t½ = 6.93 hours. The half-life calculator covers radiocarbon dating and drug-dosing examples in more depth.
Where does Euler's number come from?
Euler's number e is the limit of (1 + 1/n)^n as n approaches infinity, approximately 2.71828. It emerges naturally from continuous compounding: compounding a 100% rate infinitely often per period converges to e rather than growing without bound. The continuous growth calculator explains in more depth why e is the only base whose derivative equals itself.