How to find the growth rate of an exponential function

What Is Exponential Growth?

Exponential growth is a pattern of data that shows greater increases with passing time, creating the curve of an exponential function.

For example, suppose a population of mice rises exponentially by a factor of two every year starting with 2 in the first year, then 4 in the second year, 8 in the third year, 16 in the fourth year, and so on. The population is growing by a factor of 2 each year in this case. If mice instead give birth to four pups, you would have 4, then 16, then 64, then 256.

Exponential growth (which is multiplicative) can be contrasted with linear growth (which is additive) and with geometric growth (which is raised to a power).

Key Takeaways:

• Exponential growth is a pattern of data that shows sharper increases over time.
• In finance, compounding creates exponential returns.
• Savings accounts with a compounding interest rate can show exponential growth.

Understanding Exponential Growth

In finance, compound returns cause exponential growth. The power of compounding is one of the most powerful forces in finance. This concept allows investors to create large sums with little initial capital. Savings accounts that carry a compound interest rate are common examples of exponential growth.

Applications of Exponential Growth

Assume you deposit $1,000 in an account that earns a guaranteed 10% rate of interest. If the account carries a simple interest rate, you will earn $100 per year. The amount of interest paid will not change as long as no additional deposits are made.

If the account carries a compound interest rate, however, you will earn interest on the cumulative account total. Each year, the lender will apply the interest rate to the sum of the initial deposit, along with any interest previously paid. In the first year, the interest earned is still 10% or $100. In the second year, however, the 10% rate is applied to the new total of $1,100, yielding $110. With each subsequent year, the amount of interest paid grows, creating rapidly accelerating, or exponential, growth. After 30 years, with no other deposits required, your account would be worth $17,449.40.

The Formula for Exponential Growth

On a chart, this curve starts slowly, remains nearly flat for a time before increasing swiftly to appear almost vertical. It follows the formula:

V = S × ( 1 + R ) T V=S\times(1+R)^T V=S×(1+R)T

The current value, V, of an initial starting point subject to exponential growth, can be determined by multiplying the starting value, S, by the sum of one plus the rate of interest, R, raised to the power of T, or the number of periods that have elapsed.

Special Considerations

While exponential growth is often used in financial modeling, the reality is often more complicated. The application of exponential growth works well in the example of a savings account because the rate of interest is guaranteed and does not change over time. In most investments, this is not the case. For instance, stock market returns do not smoothly follow long-term averages each year.

Other methods of predicting long-term returns—such as the Monte Carlo simulation, which uses probability distributions to determine the likelihood of different potential outcomes—have seen increasing popularity. Exponential growth models are more useful to predict investment returns when the rate of growth is steady.

Before knowing the exponential growth formula, first, let us recall what is meant by exponential growth. In exponential growth, a quantity slowly increases in the beginning and then increases rapidly. We use the exponential growth formula in finding the population growth, finding the compound interest, and finding the doubling time. Let us understand the exponential growth formula in detail in the following section.

Meaning of Exponential Growth Formula 

Exponential growth is a pattern of data that shows an increase with the passing of time by creating a curve of an exponential function. For example, suppose a population of cockroaches rises exponentially every year starting with 3 in the first year, then 9 in the second year, 729 in the third year, 387420489 in the fourth year, and so on. The population is growing to the power of 3 each year in this case. The exponential growth formula, as its name suggests, involves exponents. There are multiple formulas involved with exponential growth models. They are:

Formula 1: f(x) = abx

Formula 2: f(x) = a (1 + r)x

Formula 3: P = P\(_0\) ek t 

Exponential Growth Formulas 

Formula 1: f(x) = abx

Formula 2: f(x) = a (1 + r)x

Formula 3: P = P\(_0\) ek t 

Where, 

• a (or) P\(_0\) = Initial value
• r = Rate of growth
• k = constant of proportionality
• x (or) t = time (time can be in years, days, (or) months, whatever you are using should be consistent throughout the problem).

Note: Here, b = 1 + r ≈ ek. In exponential growth, always b > 1.

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Examples Using Exponential Growth Formula 

Example 1: There were 50 fishes in a pond. They had increased to 135 after six months. If the fishes are growing exponentially, then how many fishes will there be in the pond at the end of one year? Round your answer to the nearest integer.

Solution:

The initial number of fishes is a = 50.

Since the fishes increased exponentially, we use the exponential growth formula.

y = a bx

y = 50 bx ... (1)

It is given that the number of fishes after 6 months is 135. So we substitute x = 1/2 (half-year) and y = 135 in the above equation.

135 = 50 (b)½

Dividing both sides by 50,

2.7 = b½

Squaring on both sides,

7.29 = b

Here, you can observe that b = 7.29 > 1, as it is exponential growth.

We have to find the number of fishes at the end of 1 year. So we substitute x = 1 and b = 7.29 in (1).

y = 50 (7.29)1 = 364.5 ≈ 365 (Rounded to the nearest integer).

Can you try this problem using any other formula of exponential growth?

Therefore, the number of fishes at the end of one year = 365.

Example 2: Jake lends $20,000 to his friend at an annual interest rate of 5.7%, compounded annually. Using the exponential growth formula, find the amount owed by his friend after 6 years? Round your answer to the nearest integer.

Solution:

The initial amount is a = $20,000.

r = rate of interest (growth) = 5.7% = 5.7/100 = 0.057.

x = number of years = 6 (we took the number of "years" here because the given rate is the "annual" rate).

By using the exponential growth formula,

f(x) = a (1 + r)x

f(x) = 20000 (1 + 0.057)6 ≈ 27,892 (Rounded to the nearest integer).

Therefore, the total amount owed after 6 years = $27,892.

Example 3: In 2001, there were 100 inhabitants in a remote town. Population has increased by 10% every year. How many residents will there be in 5 years?

Solution: 

Given, 

a = 100

r = 10% = 0.10

x = 5

Using the exponential growth formula, 

f(x) = a (1 + r)x

f(x) = 100(1 + 0.10)5

f(x) = 161.051 = 162 (rounded to the nearest integer) 

Therefore, the number of residents in 5 years will be 162. 

FAQs on Exponential Growth Formula

What is Exponential Growth Formula? 

Exponential growth is a pattern of data that shows an increase with the passing of time by creating a curve of an exponential function. We use the exponential growth formula in finding the population growth, finding the compound interest, and finding the doubling time. 

What is the Formula to Calculate the Exponential Growth? 

The formula to calculate the exponential growth is: 

f(x) = a (1 + r)x

Where, 

• a (or) P\(_0\) = Initial amount
• r = Rate of growth
• x (or) t = time (time can be in years, days, (or) months, whatever you are using should be consistent throughout the problem)

What are the Different Formulas to Calculate the Exponential Growth? 

There are two more formulas that can be used to calculate exponential growth. They are:

Formula 1: f(x) = abx

Formula 2: P = P\(_0\) ek t 

Where, 

• a (or) P\(_0\) = Initial amount
• r = Rate of growth
• k = constant of proportionality
• x (or) t = time (time can be in years, days, (or) months, whatever you are using should be consistent throughout the problem).

Note:Here, b = 1 + r ≈ ek. In exponential growth, always b > 1

If a Book Sale increased by 3% each month and got 150 sales in the First Month, how many Sales can we Expect after a Year?

Given, 

a = 150

r = 3% = 0.03

x = 1

Using the exponential growth formula, 

f(x) = a (1 + r)x

f(x) = 150(1 + 0.03)1

f(x) = 154.5= 155 (rounded to the nearest integer) 

Therefore, the sales for a year will be 155. 

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