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:
to find population growth
to find compound interest
to find doubling time
Let us understand the exponential growth formula in detail in the following section.
What is Exponential Growth Formula?
The exponential growth formula, as its name suggests, involves exponents. There are multiples formulas involved with exponential growth models. They are:
f(x) = abx
f(x) = a (1 + r)x
P = P\(_0\) ek t
In these formulas,
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.
Let us have a look at a few solved examples to understand the exponential growth formula better.
Solved 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.
To find: The number of fishes at the end of the year.
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)1/2
Dividing both sides by 50,
2.7 = b1/2
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?
Answer: 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.
To find: The total amount after 6 years.
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).
Answer: The total amount owed after 6 years = $27,892.