What Is Exponential Growth? A Complete Guide

At some point in your study of math, you’ll learn about exponential growth. Exponential growth appears in most areas of life. From the way student loan interest accrues, to the way some diseases spread, to the entire human population, exponential growth is a very important concept to understand. To answer the question, “What is exponential growth?” we will not only define its meaning, but we will also share many examples of exponential growth in application.

Get ready to put on your mathematical hat because you’re about to enter a world of numbers and equations!

What is Exponential Growth?

Exponential growth is defined as: ‘The growth of a system in which the amount being added to the system is proportional to the amount already present: the bigger the system is, the greater the increase.”

With exponential growth, the rate of growth is proportional to the number of whatever is in the system (people, organisms, money, etc.). Exponential growth shows how a quantity increases over time. If the constant is negative, that means it is exponentially decreasing. If it is positive, it is growing.

The Formula of Exponential Growth

Exponential growth is one of the most important and useful formulas in mathematics. It often only comes second in popularity to a linear function.

No matter the application, there is a formula for the function of exponential growth. It is as follows:

xt=x0(1+r)t 

Variable x is the growth at rate r, and t is time in equal intervals (as an integer). X0 equals the value of x at time 0.

On a graph, exponential growth starts off relatively flat and then curves rapidly upwards as x increases (starting flat from the left and increasing to the right).

If the graph represents exponential decay, then you will flip that curve. It will show as rapidly decreasing as the variable x increases (moving down and flattening from left to right).

Applications of Exponential Growth

To best understand how exponential growth works, it’s easiest to see it in application. There are a variety of applications, so we will cover a range from science to finance to computer science.

Let’s take a look:

Biology

Organisms and diseases spread by exponential growth. For example, an organism splits into two daughter organisms. Then, those split into two each, which is four, and the four split to form eight and so on and so forth. As you can see, with exponential growth, the amount that comes from the previous round is growing constantly. But, with more to split, more come from each round, making the growth exponential.

Additionally, in a very relevant example, the way that coronavirus is spreading is exponential. This is because when one person is infected, and as the quantity of infected people grows, so does the rate of spread. In the United States, the time it has taken for the number of coronavirus cases to double is about 2.5 days. What this means is that as time goes on, the rate of growth in cases will balloon quickly because it’s not a linear function. For example, there were the first 100,000 cases in three months. To add 100,000 more cases, it didn’t take another three months. Rather, that milestone happened in just 12 days!

When experts continued to urge for wearing masks and social distancing to “flatten the curve,” it was very much a reference to the graph of exponential growth. As you now know, exponential growth starts out flat and then shoots upwards as the numbers expand. Flattening the curve would diminish the slope (lessen the amount of new cases with each passing day).

Exponential growth of bacteria in a petri dish / https://www.pexels.com/photo/hand-holding-petri-dish-3786245/

Physics

The reason why nuclear explosions and reactions are so detrimental is because of exponential growth. The way it works is that a uranium nucleus undergoes fission. Essentially, it produces multiple neutrons which continue to get absorbed by uranium atoms nearby. Because of the rapid fission process, neutrons continue to be absorbed rather than released. The fissions happen exponentially, increasing an uncontrollable chain reaction.

Interest Rates

For college students, there are some examples of exponential growth that hit very close to home. It’s very good to understand how it works when you take out student loans. Let’s see what we mean with an example of accrued interest.

The good news is that all federal loans work off simple interest. This means that the amount you owe on the money you borrowed remains the same over time. However, some private loans may charge compound interest. That’s where exponential growth comes in.

When you take out a loan, the amount you borrow is called the principal. Interest is the cost of borrowing the money (like the additional fee you’ll owe back, generally taken as a percentage of the principal). With compound interest, you pay interest on the interest. This is because any unpaid interest is added to the principal, and then the interest is applied to that amount.

As you can see, your principal amount is essentially growing each day you don’t pay it back, so this becomes a function of exponential growth (the rate of growth grows in proportion to the amount of money owed).

When choosing loans, it’s the better financial decision to select a simple interest loan. That’ll be in your best interest (pun intended) to save you money, especially over time!

Computer Science

If you’ve studied or are interested in studying Computer Science, you’ll learn a lot about exponential growth in application. In computational complexity theory, there are some computer algorithms that require exponential resources to conduct. This could mean needing more computer memory and/or time.

For example, let’s say there’s an algorithm that needs 2x  amount of time to complete. So if x=10, then it may take 10 seconds to complete, but when x =11, it doesn’t take one more second to complete. Rather, it requires 20 seconds and the next calculation will need 40 seconds and so on. As you can see, the growth in time is not linear, as each new calculation does not take one more second to complete.

Many computers simply cannot handle this type of algorithm as they grow too quickly and require too much time and memory to process. That’s why one of the biggest goals in Computer Science is to create efficient and scalable algorithms that are solved quickly.

Savings and Finance

A lot of these examples have shown how exponential growth can be more harmful than helpful. However, exponential growth can have its upsides too. Think about saving money. Wouldn’t you rather your bank’s savings account grows rapidly rather than slowly?

The way that interest works in savings accounts of any time is also by compounding interest. For debt, it will be hurtful to the borrower. But, if you flip the equation and apply it to your savings, then compounding interest is a good thing! This is the reason why most people will stress the importance of saving for retirement early on.

When putting away money for retirement and starting early, the exponential growth will work in your favor. If we use the example of putting away $1,000 today, we can see how much value it’ll create with just a .1% interest rate. After 10 years, you’ll have more than $16,000, but after 20 years, this will grow to over $116,000!

Exponential growth shown in a dashboard of website analytics / https://www.pexels.com/photo/apple-devices-books-business-coffee-572056/

Marketing

You can also see how exponential growth works to help businesses thrive. In marketing, there’s a concept of SEO, or search engine optimization. In essence, search engines like Google rank your business’ website and content. People use search engines by putting in keywords of what they’re looking for. If the keyword matches your product or service, then your website may pop up. But how does Google choose what to show as the first results on the page?

Search engines work off their own algorithms. To boost your likelihood of ranking high on Google, companies use marketers and copywriters to boost their SEO. But this isn’t something that is a quick and overnight fix. Instead, it takes time to ramp up and start building. Once it takes off, it has the power to grow exponentially. This is because a website’s traffic will grow naturally when it appears as a result high up on the list on Google. With more people clicking on it, the higher it can continue to rank.

With better SEO should hopefully come higher sales and business growth. Unlike biological processes or finance where the output can easily be calculated and tied to the input, marketing and business’ exponential growth may seem a bit more challenging. Yet, with today’s amount of data and tools to process data, business executives have better ways to assess their return on investment (ROI). This helps them know where to allocate money to see the best results to grow their business.

Doubling Time

With exponential functions, there’s a rule on how to calculate its doubling time. Doubling time is the time it takes for the amount to double (sounds straightforward enough, right?). The Rule of 72 states that an exponential function will double at: 72/growth rate.

Let’s pretend we have an investment with an interest rate of 5%. That means that your initial investment will double every 14.4 years (72/5). If you start with $100, then you will add $5 for $105 going into the next year. Then, you will receive 5% of $105, which is $5.25, so you will then have $110.25. 5% of $110.25 is $5.51. As you can see, the interest is growing every time your total investment grows.

The Rule of 72 can also help you see how big of a difference just 1% in interest can cause on your return on investment. The doubling rate can move from 14.4 years to 12 years when you go from 5% to 6% interest.

The Wrap Up

Exponential growth shows how growth (or decay) occur proportionally to what’s being added to the system. With some things, exponential growth can prove to be very beneficial. This is the case with the growth of your money in a savings account or the growth of your business, thanks to a successful marketing campaign.

On the other hand, the uncontrollable nature by which some processes occur can cause negative effects because of exponential growth. The world has seen this with pandemics like the wide and rapid spread of coronavirus. The same negative effects can be seen with nuclear weapons, because the impact is so much more widespread and long-standing than many humans can even comprehend.

When it comes to being a student, understanding exponential growth can play a role in how you select your student loans. As mentioned, all federal aid will work on simple interest. But, if you come across a loan with compound interest, it’s best to avoid it if you have the option. Beyond student loans, you can consider applying for scholarships, grants or attending an affordable online university like University of the People.

Exponential growth is a function of math that will continue to exist in many aspects of life. From finance, biology, computer science, business, physics, and more, it’s a mathematical function that can be applied to so much.

Since the numbers can get so large so quickly, it may be hard to fully grasp exponential growth. But, if you take just one thing away from this list of examples, it’s that growth happens in proportion to what’s already there.

An easy way to think about exponential growth is through social media challenges, or pyramid schemes. If you challenge three people to post a photo, and each person has to do the same, then once you’ve started, you’ve already grown the challenge to 9 people on it’s second round. Each person of 9 challenges three more, and now you’ve quickly reached 18. Exponential growth is the reason why things go viral so fast!

Exponential growth is a function filled with power (that can quickly become uncontrollable). That’s why understanding the implications of exponential growth is so important for every person.