Exponential Growth and Decay Formula
The exponential growth and decay formula are used to tell whether the exponential function will grow with time or decay with time. For any exponential function, the basic equation used is:
\(y = y_0 b^{kt}\)
Here, b is known as the base of the function, k is the growth rate, y_{0} is the initial value of the function. So, whether the function will exponentially grow or decay, depends on the value of k. If k>0, the function will be an exponential growth function, and if the k<0, then the function is an exponential decay function. The exponential growth and decay formula are explained below along with solved examples.
What are the exponential growth and decay formula?
The formula for exponential growth and decay is given as,
\[ \text {Exponential Growth} \implies y = y_0 b^{kt}\]
\[ \text{Exponential Decay} \implies y = y_0 b^{kt}\]
The curves for exponential growth and decay functions can be seen below:
Solved Examples Using Exponential Growth and Decay Formula

Example 1:
Calculate the population growth of coronavirus in 5 hours, if the growth is given by the function: f(t) = 200e^{0.02t}. Here t is represented in minutes.
Solution:
To find: Population of coronavirus
Given, t = 5hr = 300 minutesPutting the value of t = 300 minutes in the function.
f(300) = 200e^{0.02(300) }≈ 80,686.
Answer: The population of coronavirus is 80686 after 5 hrs.

Example 2:
If the population of bacteria decreases as f(t) = 500e^{0.04t}, after putting antibacterial fluid, where t is in minutes. Find the number of bacteria left after 40 minutes.
Solution:
To find: Population of bacteria
Given, t = 40 minutesPutting the value of t in the function.
f(40) = 500e^{0.04(40) }≈ 100.95.
Answer: The population of bacteria after 40 minutes is 100.95.