Logarithmic and Exponential Growth Computer

Forecast scale schedules with continuous, discrete, and logistic growth models. Live SVG curve, 10-point schedule table, and doubling time - all computed locally in your browser.

Growth Forecaster

Enable logistic mode via the preset or by setting K above N0

Final Projected Value
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Doubling Time (Td)
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Absolute Growth
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Growth Curve Carrying Capacity (K)
10-Point Forecast Schedule
Milestone Time (t) Value N(t) Growth from N0 Growth %
Key Terms Explained
Exponential Growth
A pattern where the value multiplies by a constant factor each period, producing a J-shaped curve that accelerates without bound (absent a limit).
Logarithmic Scale
A measurement axis where each tick mark represents a power of 10 (or another base) rather than an equal increment. It compresses wide ranges so exponential data appears as a straight line.
Logistic Growth
An S-shaped growth pattern that starts exponential but decelerates as the population approaches its carrying capacity, creating a natural ceiling.
Carrying Capacity (K)
The maximum sustainable population or value in a given environment or market. In logistic models, growth slows and asymptotes toward K.
Doubling Time (Td)
The number of periods required for a value to double. For continuous growth: Td = ln(2) / r. For discrete growth: Td = ln(2) / ln(1+r). Shortcut: 72 divided by the rate %.
Base e / Euler's Number
The irrational constant approximately equal to 2.71828. It is the natural base for continuous growth because the derivative of e^x equals e^x itself.
Asymptote
A line that a curve approaches but never quite reaches. In logistic growth, K is the horizontal asymptote that limits the maximum value.
Compounding
Earning returns on previously earned returns. Discrete compounding adds interest at set intervals; continuous compounding does so at every instant, maximizing the effective yield.

The Complete Guide to Exponential and Logistic Growth Forecasting

Whether you are modeling a startup's user growth, a portfolio's compound returns, or an epidemic's spread, the underlying math is the same: a value grows in proportion to its current size. This guide explains the three core formulas, when to use each, and how to avoid common mistakes when projecting at scale.

How to Use This Tool

Select a preset to auto-fill realistic starting values, or choose "Custom Forecast" to enter your own. Use the Growth Type toggle to switch between continuous (e^rt) and discrete ((1+r)^t) compounding. Enable the logistic limit by entering a Carrying Capacity (K) larger than the Initial Value, which forces the model into S-curve mode. All outputs - the hero metrics, the SVG curve, and the 10-point schedule table - update instantly as you move any slider.

Discrete vs. Continuous: Which Should You Use?

Use discrete compounding when interest or returns are credited at defined intervals - annual salary raises, monthly bank statements, or quarterly dividends. Use continuous when modeling biological populations, viral app adoption, or any process where growth happens at every moment rather than at fixed checkpoints. Continuous growth always exceeds discrete at the same nominal rate because compounding never pauses.

The Logistic Model and Market Saturation

Pure exponential models eventually predict absurd outcomes - a social network cannot have more users than people exist on Earth. The logistic formula corrects this by incorporating carrying capacity K. Early in the S-curve, growth mimics exponential. As the value crosses roughly K/2, the growth rate peaks. Beyond that point, deceleration mirrors the acceleration from the early phase, and the curve asymptotes toward K. This is why viral product adoption, epidemic spread, and market share all produce S-curves in the real world.

Handling Very Large Numbers

Exponential math scales violently. A 20% annual rate compounding for 50 years turns $1,000 into over $9 million. Rates above 50% over long periods quickly exceed JavaScript's safe integer range (approximately 9 quadrillion). This tool automatically switches to scientific notation when values exceed 1 billion, and caps the chart's Y-axis to prevent rendering issues. If your inputs produce scientific notation output, consider whether the logistic model with a realistic K is more appropriate.

Rule of 72 Cheat Sheet

The Rule of 72 is a mental math shortcut: divide 72 by the annual percentage rate to get the approximate doubling time in years. No calculator needed. It is accurate to within 1-2% for rates between 3% and 15%.

Formula: Td = 72 / r%

Rate (%)Approx. Doubling TimeExact Doubling Time
2%36 years35.0 years
4%18 years17.7 years
6%12 years11.9 years
8%9 years9.0 years
10%7.2 years7.3 years
12%6 years6.1 years
18%4 years4.2 years
24%3 years3.2 years
36%2 years2.3 years
72%1 year1.3 years

Frequently Asked Questions

Discrete growth compounds at fixed intervals using N(t) = N0 x (1+r)^t, adding a percentage each period on top of the running total. Continuous growth compounds at every instant using N(t) = N0 x e^(rt), where e is Euler's number (approximately 2.71828). Continuous always produces a slightly higher result for the same nominal rate because compounding never pauses between intervals. Banks advertise APR (discrete) but pay APY, which reflects continuous or frequent compounding. In biology and viral adoption, continuous is the more accurate model because organisms and networks grow at all times, not just at year-end.
Euler's number e (approximately 2.71828) is the unique base at which the rate of growth of a function equals the function itself. If N(t) = e^t, the slope of the curve at any point exactly equals its height at that point. This property makes e the natural base for any process where the rate of change is proportional to the current value, which is the defining characteristic of continuous exponential growth. It was discovered by Jacob Bernoulli while studying compound interest and formalized by Leonhard Euler.
Carrying capacity (K) is the maximum population or value that an environment or market can sustainably support. In the logistic formula N(t) = K / (1 + ((K - N0) / N0) x e^(-rt)), the curve starts exponential but decelerates as it approaches K, eventually leveling off into a flat asymptote. Real examples: a social network's user base approaching total internet users, a bacteria colony limited by nutrients, or a new product saturating its target market. Without a carrying capacity, pure exponential models eventually produce unrealistic results.
For continuous growth: Td = ln(2) / r, where ln(2) is approximately 0.6931 and r is the decimal growth rate. For discrete growth: Td = ln(2) / ln(1+r). The Rule of 72 mental shortcut says to divide 72 by the percentage rate to get the approximate doubling time. A 6% annual return doubles in about 72 / 6 = 12 years. This is accurate to within 1-2% for rates between 3% and 15%, making it a fast sanity check without a calculator.
Viral trends follow a logistic S-curve because every population has a finite ceiling. Early adoption is exponential: each new user can spread to many people who have not yet heard of the product. But as adoption rises, the pool of reachable non-users shrinks. Each new user has fewer potential converts to reach, so the growth rate decelerates. Eventually nearly everyone who would adopt the product already has, and growth flatlines at the carrying capacity. The same pattern governs epidemic spread, technology diffusion, and market saturation across industries.
This tool is for educational and planning purposes only. Growth projections are mathematical models, not guaranteed outcomes. Real-world results depend on market conditions, competition, and many factors this model cannot capture. Not financial advice.