What is exponential growth? (Definition and examples)

Updated 12 April 2023

Exponential growth occurs when an original amount or quantity increases at a consistent rate over a period of time. Analysts use this mathematical concept to make predictions about the future, such as the value of an upcoming investment, the trajectory of an infectious disease or the expected population of a city. It's important to understand the significance of exponential growth and the way it impacts our day-to-day lives. In this article, we define exponential growth and explore various real-world examples of this phenomenon.

What is exponential growth?

Exponential growth refers to a pattern of data that shows steep and consistent increases over time. The consistent doubling or tripling of a quantity over a fixed period, such as over an hour, a day or a week often signifies rapid growth. Although this type of growth often starts slowly and may appear insignificant in its early stages, the numbers increase rapidly. This can have positive, negative or even disastrous consequences, depending on the situation.

Rapid growth is also an important concept, which is why many industries and fields use it. These include biologists, scientists, financial analysts, mathematicians, economists and business leaders to conduct research and development projects around the world. It's essential to recognise rapid growth when it occurs and to understand its effects.

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How does exponential growth work?

To understand how exponential growth works, it's helpful to review real-life situations that may occur.

Example: A vet researcher is tracking how a controlled population of dogs increase from year to year and notices it doubles every year. In the first year, they start with two dogs, and by year two, they have four dogs. So, in year three, the number increases to eight dogs, and in year four, 16 dogs. The number steadily increases by two each year, which helps the researcher track the growth pattern.

This represents rapid growth because the numbers not only increase, they increase at a consistent rate. In theory, exponential growth can continue indefinitely. In practice, it usually comes to a natural end after a period of time due to external or environmental factors. The opposite of exponential growth is exponential decay. With exponential decay, you reduce an amount by a consistent percentage over time. The numbers also reduce quickly and rapidly.

Exponential growth vs linear growth

Exponential growth is different from other growth measurements, such as linear growth:

Exponential growth

Exponential growth is multiplicative and involves repeated multiplication. Here's an example to illustrate this.

Example: A shop owner uses a word-of-mouth marketing campaign to increase the number of consumers who visit the shop. At the start of the campaign,10 consumers visited the shop, and they each recommended the shop to five people. With a repeated word-of-mouth campaign such as this, from week to week, the shop gains five times the amount from the previous week.

If the campaign is successful and the number of consumers who visit the shop, each week grows at an exponential rate, the shop gains:

Week 0: 10
Week 1: 50
Week 2: 250
Week 3: 1,250
Week 4: 6,250

Linear growth

Linear growth is additive and grows by the same amount over time. For example, if the shop owner increases the number of consumers who visit the shop by five each week, it demonstrates linear growth. With this pattern of growth, the shop owner gets this result:

Week 0: 10
Week 1: 15
Week 2: 20
Week 3: 25
Week 4: 30

Formula for exponential growth

For more complex situations, you can use a formula to calculate rapid growth:

V=S×(1+R)T

In this formula:

V = the current value of the quantity or amount that's subject to exponential growth
S = the initial starting value of the quantity or amount that's subject to exponential growth
R = the rate of growth, expressed as a decimal
T = the number of time intervals

You can use this formula to plot rapid growth on a graph. When you do so, it tends to result in a curve that starts slowly and remains flat for a time, before increasing rapidly and becoming almost vertical. The size of the R value (the rate of growth) in the formula for exponential growth determines how rapidly the curve rises.

Real-world examples of rapid growth

Rapid growth isn't an abstract theory. It has many real-world applications, and it affects many aspects of life. Here are some examples:

Compound interest

A savings account with a compounding interest rate can show exponential returns over time. It allows you to invest a small amount of money and transform it into a significant sum. With compounding, you earn interest on the cumulative total in the account each year. Suppose you invest €100 at a compound interest rate of 10%. At the end of the first year, you earn €10. At the end of the second year, the bank applies the 10% interest rate to the cumulative total in the account, which is now €110. As a result, you earn €11 in interest.

As each year passes, you earn a higher amount of interest. This soon results in rapid growth. Financial modellers can use exponential growth to forecast the future financial value of investments, although it's less helpful in situations where the interest rate fluctuates.

Bacteria

Bacteria grow and multiply exponentially and typically take roughly an hour to reproduce through a process called binary fission. With this process, each bacterium cell divides into two daughter cells, which further divide into two cells each. If you place bacteria in the right conditions and observe their numbers, they rapidly display exponential growth. Over the course of a day, one single bacterium can generate millions of new bacteria. Eventually, the bacteria run out of food and nutrients. At that point, the growth rate slows and levels off.

Viruses

Viruses can spread rapidly, and typically one person transfers and spreads it to the people they come in close contact with, such as colleagues, friends and family members. This increases the number of people infected with the virus. They, in turn, go on to infect other people. If the virus is transmissible enough, the numbers affected can quickly show exponential growth. This may even result in a global pandemic.

Between 1918 and 1919, influenza spread rapidly and exponentially around the world. In a short time, this virus infected about one-third of the world's population. In more recent times, the COVID-19 coronavirus also spread exponentially, with over 500 million confirmed cases recorded globally.

Internet memes

People around the world use the Internet to share all kinds of information, from recipes and product recommendations to jokes and funny memes. On social media channels and digital platforms, you can quickly and easily share information with friends and followers by clicking a button. The information can reach hundreds or even thousands of people in an instant. These people can forward the information to their own friends and followers, who may continue the process. In this way, the spread of memes and other data over the Internet is exponential.

Fire

The spread of fire also displays this type of growth. There's an exponential relationship between the fire and the items it ignites. When an object catches fire, it transfers its energy to the objects in its immediate surroundings and sets them on fire too, assuming they're combustible. Those objects go on to ignite the objects that surround them. As the cycle continues, the number of objects on fire multiplies and the fire spread at a rapid and exponential rate.

Water lilies

Water lilies grow and reproduce in a way that follows the exponential growth curve. They spread slowly at first. After a while, the rate of spread speeds up and they can eventually take over an entire pond, lake or body of water.

Pyramid schemes

A pyramid scheme is an illegal investment vehicle that relies on the concept of exponential growth to make a profit. The operators of these schemes initially recruit a number of investors, who recruit more investors and so on. The schemes tend to grow exponentially, but then run out of new recruits and collapse. This often results in investors losing their money.

What is exponential growth? (Definition and examples)