Exponential Growth

Before answering what is exponential growth? Let us know, why is it called exponent? An exponent refers to the number of times a number is multiplied to itself. We can have a clear idea on the topic by solving exponential growth equations.

Something which always grows in relation to the current value is known as exponential growth. Exponential growth is also known as doubling the existing number. Let us take an example:  If the population of rabbits grows every month, then we would have 2, then 4, 8, 16, 32, 64, 128, 256, and further carried on. The formula used in solving exponential growth equations is y = ab^x.

Exponential Growth Graph

The general shape of an exponential growth graph is ‘J’ shape. This shows that the graph is always increasing in nature. An exponential growth graph is drawn using the function of the form y = ab^x where, a > 0 and b > 1. Note that the graph shoots upward rapidly as ‘X’ increases. This is because of the doubling behavior of the exponential. Below given graph is an example of exponential growth.

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Exponential Growth Equation

For solving exponential growth equations you need to have equations with an exponential expression on both sides of the equal sign. Then compare the powers and solve the given equation. Let us solve one basic problem to understand it better.

Q. Solve  5^1-x = 5⁴

As the bases are the same we can equate the powers and solve the problem.

1 - x = 4

x = -3

This is the required solution to the above problem.

It is used in solving exponential growth equations when there is a comparison between two cases.

Exponential Decay

Something which ‘Decay’(gets smaller) with respect to time exponentially is known as exponential decay. Exponential decay is applied when anything decreases compared to the current value exponentially. For example, we can take atmospheric pressure of the air around us which decreases as we go up. It decreases by about 12% for every 1000m which is exponential decay.

Exponential Decay Formula

The formula used to express the exponential decay is  y = a(1 - b)^x.

Where y is the final amount, a is the original amount, b is the decay factor, and x the amount of time that has passed. Exponential decay formula is useful in solving a variety of real-world problems, Most popular application is to track inventory that is used regularly in the same quantities.

Solved Problems

Q. The population of a city in 2016 estimated to be 35,000 people with an annual rate of increase of 2.4%.

Solution:  Now let us find the growth factor,

After one year population would be = 35,000 + 0.024(35,000)

By factoring above equation becomes = 35,000 (1 + 0.024)

The growth factor is b = 1.024 (Remember that it is greater than 1)

Now, The general formula for exponential growth is y = ab^x

Substituting the value in above formula  y = 35,000(1.024)^x

Consider, that we are using this estimate of the population in 2020 to the nearest hundred people.

Y = 35,000 (1.024)4≈ 38,482.91 ≈ 38,500

So, the estimated population of the city in 2020  is 38,500.