What is exponential growth? Why doubling surprises people

Exponential growth is growth in proportion to the current size: the bigger something is, the faster it grows, so it increases by the same percentage in each period rather than by the same amount. A quantity growing 10% a year adds 10 in its first year from 100, but about 24 in its tenth. Its defining sign is a fixed doubling time, and it is why such growth looks slow for a long while and then sudden.

In short

• Linear growth adds the same amount each step; exponential growth multiplies by the same factor each step.
• Anything growing exponentially doubles at regular intervals, and the rule of 70 estimates the interval: 70 divided by the percentage growth rate.
• No real quantity grows exponentially for long; limits bend the curve into an S shape.

How is exponential growth different from linear growth?

Take two quantities that both start at 100. One grows linearly, adding 10 each step. The other grows exponentially, adding 10% of whatever it is at that moment. After one step both are at 110, and for the first few steps they are hard to tell apart.

Step	Linear: add 10	Exponential: add 10%
0	100	100
1	110	110
2	120	121
3	130	133.1
4	140	146.4
5	150	161.1
6	160	177.2
7	170	194.9
8	180	214.4
9	190	235.8
10	200	259.4
20	300	672.7
30	400	1,744.9

The difference is in what each step adds. The linear column always adds 10. The exponential column adds 10, then 11, then 12.1, and by step 30 it is adding more than 150 a step. Exponential growth feeds on itself: each step's growth becomes part of what grows next. Money earning compound interest works exactly like the right-hand column, and so does a price level under steady inflation.

How do you write exponential growth down?

The formula is N = N₀ × (1 + r)^t. Here N₀ is the starting amount, r is the growth rate per period written as a decimal (10% is 0.1), t is the number of periods, and N is the amount after t periods. The variable t sits in the exponent, the small raised number that says how many times to multiply, which is where the name comes from. For the table above, step 10 is 100 × 1.1^10, which is about 259.4.

The same growth can be written with its doubling time T instead: N = N₀ × 2^(t/T). Every time t grows by T, the amount doubles. Seen as a function of time, an exponential curve lies flat along the axis at first and then turns steeply upward. Plotted on a logarithmic scale, where each step up the axis means ten times more, the same curve becomes a straight line, which is how scientists spot exponential growth in data. Working backward from an amount to the time it took needs a logarithm.

How long does it take to double?

A handy shortcut, the rule of 70, says that a quantity growing at p percent per period doubles in about 70 ÷ p periods. Something growing 7% a year doubles in about 10 years; at 2% a year, about 35. The number 70 comes from the natural logarithm of 2, which is 0.693. Some people use 72 instead, because it divides evenly by more numbers. Here is how good the shortcut is:

Growth per period	Rule of 70	Exact doubling time
1%	70	69.7
2%	35	35.0
5%	14	14.2
7%	10	10.2
10%	7	7.3
20%	3.5	3.8

It is close for small rates and drifts as the rate gets large. The useful lesson is the reverse: anything with a steady doubling time is growing exponentially, whatever is causing it.

Why does doubling surprise people?

The old legend of the chessboard shows it. A ruler offers a reward, and the inventor of chess asks for one grain of wheat on the first square, two on the second, four on the third, doubling on each of the 64 squares. It sounds modest. The last square alone holds 2^63 grains, and the whole board holds 2^64 − 1, which is 18,446,744,073,709,551,615 grains, a number with 20 digits. At about 65 milligrams a grain, that is roughly 1.2 trillion tonnes of wheat, more than 1,400 times the world's whole wheat harvest of 2025–26. The story was first written down by the historian Ibn Khallikan in 1256.

Two features of the story catch people out. First, the early squares are deceptive: after the first row of 8 squares, the whole pile is only 255 grains. Second, each square holds one more grain than all the squares before it put together. So the last stretch dwarfs everything earlier: the whole first half of the board, 32 squares, holds about 4.3 billion grains, roughly 279 tonnes, while the second half holds more than 4 billion times as much. A pond weed that doubles daily and covers the pond on day 30 covered only half of it on day 29, and less than a thousandth on day 20.

People tend to judge growth by extending a straight line through what they have seen so far. With exponential growth that guess is too low by a margin that widens at every step.

Why does exponential growth always stop?

Exponential growth needs its cause to stay the same, and in the real world it doesn't. Bacteria such as E. coli can divide about every 20 minutes in ideal laboratory conditions. Kept up for one day, that would be 72 doublings, about 4.7 × 10²¹ cells from one, far more than any flask could hold. In practice the food runs out and waste builds up, and the growth slows and stops.

The shape that results is called logistic growth, an S-shaped curve. It starts out exponential, slows as it nears a limit, called the carrying capacity, and levels off. The two curves are almost identical at the start, which is why an early run of doubling says little about where the growth will end.

Epidemics follow the same pattern. In the first weeks of COVID-19 in Wuhan, a study of the first 425 confirmed cases, reported by 22 January 2020, estimated that in its early stages the epidemic doubled in size every 7.4 days (Li and others, New England Journal of Medicine, 2020). Doubling every week turns 100 cases into about 13,000 in seven weeks. Growth like that slows when people change their behavior, when public-health measures cut contacts, or when enough people become immune that each case infects fewer others. Early exponential growth is a warning about what happens if nothing changes, not a forecast.

What do people get wrong about exponential growth?

The most common mistake is to use "exponential" to mean "very fast". It means growth at a steady rate in proportion to size, which can be slow: 1% a year is exponential, and it takes about 70 years to double. A rocket climbing at a steady 1,000 meters a second is fast but linear. The other mistake runs the opposite way: assuming a doubling trend will carry on forever, when every real one meets a limit.

The same arithmetic also works in reverse. Exponential decay shrinks by the same fraction each period, and its version of the doubling time is the half-life, the idea behind radioactive decay and how drugs leave the body. When growth is compounded continuously rather than in steps, the base of the exponent becomes a special number, e, about 2.718.

Check yourself

• Without looking back, explain why a quantity growing 10% a step adds more at step 30 than at step 1, while a linear one adds the same.
• Using the rule of 70, about how long does something growing 5% a year take to double, and to quadruple?
• Why does a real epidemic or bacterial colony stop growing exponentially, and what shape does its curve take instead?

Sources

• Early Transmission Dynamics in Wuhan, China, of Novel Coronavirus–Infected Pneumonia — Qun Li and others, New England Journal of Medicine, 2020
• Wheat and chessboard problem — Wikipedia
• Logistic function — Wikipedia
• Exponential and logarithmic functions — OpenStax, Calculus Volume 1, Rice University, 2016
• Exponential growth — Wikipedia