Exponential Growth & Decay
Drag along the curve and watch the tangent slope equal the height — the defining property of eˣ. Growth, decay and why e matters, made visual.
Bacteria in a dish, money in a savings account, a radioactive sample, a viral rumour — whenever the rate of change depends on how much is already there, you get exponential behaviour: the runaway curve that doubles, and doubles, and doubles again.
Growth that feeds on itself
A quantity grows exponentially when its rate of growth is proportional to its current size. The more there is, the faster it grows — so it keeps doubling over equal time intervals. Flip the sign and you get exponential decay, which halves over equal intervals instead.
The standard model is N = N₀eᵏᵗ, where N₀ is the starting amount, t is time, and k is the growth constant — positive for growth, negative for decay. That constant multiplier is the whole point: exponential change multiplies, it does not add.
Interest earns interest; the bigger it gets, the faster it grows.
See it: slopes and doubling
Switch between the curves below and drag the point along each. For eˣ watch the tangent: its slope exactly equals the height of the curve at every position. Then compare the falling decay curve e⁻ˣ against the doubling curve 2ˣ.
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A graph of exponential functions with a draggable point and tangent. For eˣ the slope of the tangent always equals the height of the curve — the defining property that makes e the natural base. The decay curve e⁻ˣ falls towards zero, halving over equal intervals, while 2ˣ doubles over equal intervals.
Why e is the natural base
You could write exponential growth with any base — 2ᵗ, 10ᵗ, whatever suits. So why does e ≈ 2.718 get called the natural base? Because of one beautiful property: the function eˣ is its own derivative.
Choose any other base and a stray constant (a natural log) tags along when you differentiate. Base e is the one that makes it vanish, so e is the base that Nature — and every exam mark scheme — prefers.
Worked examples
A culture starts with 500 cells and doubles every hour. Write a model for the number of cells N after t hours, and find the population after 3 hours.
Doubling each hour means multiplying by 2 every step, so N = 500 × 2ᵗ. After 3 hours: N = 500 × 2³ = 500 × 8 = 4000 cells. Notice how the same base-2 idea can also be written with base e as N = 500eᵏᵗ where k = ln 2 ≈ 0.693 — the two forms describe the identical curve.
The same shape drives continuous compound interest: money invested at a continuous rate follows N = N₀eᵏᵗ, growing by a fixed percentage of its current value in every instant rather than a fixed number of pounds.
Common mistakes
Practice
A population grows by 5% every year. Is that linear or exponential growth, and what is the growth factor per year?
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It is exponential: a fixed percentage means multiplying by the same factor each year, not adding a fixed amount. The growth factor is 1 + 0.05 = ×1.05 per year, so after t years the population is N₀ × 1.05ᵗ.
Where next?
The inverse question — “how long until it doubles, or halves?” — is answered with logarithms, the natural companion topic that unpicks an exponent to solve for time.
Frequently asked questions
What is exponential growth?+
Why is eˣ special?+
What is the difference between exponential growth and decay?+
How do you differentiate an exponential function?+
The ScholarsGate Maths Team
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Written and reviewed by ScholarsGate tutors who teach A-Level and undergraduate mathematics. Every explainer is checked for accuracy against the AQA, Edexcel and OCR specifications.
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