109 LearnersLast updated on July 9th, 2025
Exponential Growth and Decay
Exponential growth and decay describe how quantities increase or decrease quickly over time, such as the growth of a tree (increase) or decline in the stock market (decrease). The increase is represented by exponential growth and the decrease by exponential decay.
What are Exponential Growth and Decay?
Physical quantities experience changes over time, and they can be studied using exponential growth and decay. When the change is not constant but exponential, it is termed exponential growth or exponential decay. Exponential growth tells us how something increases with time, and exponential decay shows how something reduces with time. It is expressed in the form f(x) = abx, where
a = initial quantity
b = growth factor
and x is the time over which the growth and decay happen.
If the value of b >1, it represents exponential growth, and if b< 1, it represents exponential decay.
What are the Formulas for Exponential Growth and Decay
Now, let’s learn how to calculate exponential growth and decay. The formulas to calculate them are given below:
|
Exponential Growth |
Exponential Decay |
|
f(x) = abx |
f(x) = ab-x |
|
f(x) = a(1 + r)t |
f(x) = a(1 - r)t |
|
P = P0ekt |
P = P0e-kt |
Where a or P0 represents the initial quantity of the substance, r represents the rate of growth, and t represents the time steps.
Real-World Applications of Exponential Growth and Decay
Exponential growth and decay have many real-life applications. They are used in various fields as mentioned below:
- In chemistry, we use exponential decay in the concentration of reactants over time to study first-order reactions. This helps us understand the reaction speeds in various chemical processes.
- Exponential growth and decay are used to analyze viral or bacterial growth. This is an important application because it helps us understand the spread of various diseases.
- While analyzing feedback from customers, the number of responses can grow exponentially.
Common Mistakes and How to Avoid Them in Exponential Growth and Decay
Exponential growth and decay are the basis of many fields, from population dynamics to radioactive decay analysis. In this topic, we learned more about exponential growth and decay.
Mistake 1
Confusing Growth and Decay
Students tend to think that all exponential functions represent growth without checking the sign of the exponent. So it is important to check whether ‘r’ is being subtracted or added to 1. If the form (1 + r) represents growth, and the form (1 - r) represents decay.
Mistake 2
Forgetting the Initial Value
Not identifying or forgetting the initial value can result in errors even if the rate is correct. So, first identify what P0 is and what quantity is at t = 0.
Mistake 3
Misunderstanding Continuous Growth
Students tend to confuse continuous compounding with periodic compounding, and they also use the wrong formulas. So, read the question carefully before deciding if the process is continuous or discrete.
Mistake 4
Incorrect Time Units
Students make silly mistakes in calculating the time units. If the growth rate ‘r’ is per year, then it should also be in years. Using mismatched units can lead to errors in calculations. So always both rate and time should be in the same units.
Mistake 5
Thinking Negative Exponents Always Mean Decay
Students often think that negative exponents always mean decay; that is not necessarily true. A negative exponent may cause decay, but not all negative exponents imply decay; make sure to understand the entire expression and do not take the negative exponent out of context.
Solved Examples of Exponential Growth and Decay
Problem 1
A city has a population of 50,000 people, and it grows at a rate of 3% per year. Find the population after 5 years.
After 5 years, the population is approximately 58,090
Explanation
To calculate the population growth, we use the formula P = P0ekt
Where, P0 = 50,000
k = 3% = 0.03
t = 5 years
So, P(5) = 50,000 × e(0.03 × 5)
= 50,000 × e0.15
= 50,000 × 1.1618 = 58,090
Problem 2
An investment of $1,000 is placed in a bank that offers a 5% annual interest rate, compounded continuously. Find the amount after 10 years.
The investment after 10 years is equal to $1648.72
Explanation
For continuous compounding, the amount after t years is calculated using;
A = P × ekt
Here, P = $1000
k = 5% = 0.05
t = 10 years
e = 2.718
So, A = 1000 × e0.05 × 10
As, e0.5 = 1.64872
So, A = 1000 × 1.64872 = 1648.72
Problem 3
A bacterial culture starts at 100 mg and grows by 7% per hour. Find the amount after 8 hours.
The amount increase in bacterial culture after 8 hours is 171.51 mg
Explanation
The exponential growth can be calculated by using the following formula;
A = P0(1 + r)t
Here, P0 = 100 mg
r = 7% = 0.07
t = 8 hours
So, A = 100 (1 + 0.07)8
= 100 × 1.078
= 171.8 mg
Problem 4
A radioactive substance has 200 grams and decays at a rate of 5% per year. Find the remaining amount after 6 years.
The radioactive decay after 6 years is 143.26 grams
Explanation
The exponential decay can be calculated by
A = P0(1-r)t
Here, P0 = 200 grams
r = 5% = 0.05
t = 6 years
So, A = 200 (1 - 0.05)6
= 200(0.95)6
= 143.26 grams
Problem 5
A social media account has 500 followers and grows at a rate of 6% per week. How many followers will it have after 10 weeks?
The growth after 10 weeks is approximately 895 followers
Explanation
The exponential growth is calculated using f(x) = a(1 + r)t
Where, a = 500
r = 6% = 0.06
t = 10 weeks
So, f(x) = 500(1 + 0.06)10
= 500 × 1.0610
= 500 × 1.7908 = 895.4
FAQs on Exponential Growth and Decay
1.What is exponential growth?
2.What is exponential decay?
3.What is the formula for exponential growth?
4.What is the formula for exponential decay?
5.What are some real-world applications of exponential growth and decay?
6.How can children in Australia use numbers in everyday life to understand Exponential Growth and Decay?
7.What are some fun ways kids in Australia can practice Exponential Growth and Decay with numbers?
8.What role do numbers and Exponential Growth and Decay play in helping children in Australia develop problem-solving skills?
9.How can families in Australia create number-rich environments to improve Exponential Growth and Decay skills?
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Hiralee Lalitkumar Makwana
About the Author
Hiralee Lalitkumar Makwana has almost two years of teaching experience. She is a number ninja as she loves numbers. Her interest in numbers can be seen in the way she cracks math puzzles and hidden patterns.
Fun Fact
: She loves to read number jokes and games.
