What is exponential decay?

What is exponential decay? Whenever something is decreasing or shrinking rapidly as a result of a constant rate of decay applied to it, that thing is experiencing exponential decay.

The figure below is an example of exponential decay. In fact, it is the graph of the exponential function y = 0.5^x

The general form of an exponential function is y = ab^x. Therefore, when y = 0.5^x, a = 1 and b = 0.5.

The following table shows some points that you could have used to graph this exponential decay. Try to locate some of these points on the graph!

x	2^x	y
1	0.5¹	0.5
2	0.5²	0.25
3	0.5³	0.125
0	0.5⁰	1
-1	0.5^-1	1/0.5 = 2
-2	0.5^-2	1/0.25 = 4
-3	0.5^-3	1/0.125 = 8

What is exponential decay in real-life?

There are many real-life examples of exponential decay. For example, suppose that the population of a city was 100,000 in 1980. Then every year after that, the population has decreased by 3% as a result of heavy pollution. This is an example of exponential decay.

Notice that the rate of decay is 1% or 0.01 and it is constant. This is important since the rate of decay cannot change.

Let us find the exponential function.

Year 1981 or 1 year after:

100,000 - 100,000 x 0.03 = 100,000 (1 - 0.03 )

100,000 - 100,000 x 0.03 = 100,000(0.97)

100,000 - 100,000 x 0.03 = 100,000(0.97)¹

Year 1982 or 2 years after:

100,000(0.97) - 100,000(0.97) x 0.03 = 100,000(0.97) [1 - 0.03]

100,000(0.97) - 100,000(0.97) x 0.03 = 100,000(0.97)(0.97)

100,000(0.97) - 100,000(0.97) x 0.03 = 100,000(0.97)²

Following this pattern, suppose that

x is the number of years since 1980
100,000 is the starting amount
0.97 is the rate or decay factor

Then y = 100,000(0.97)^x

Comparing this exponential function with y = ab^x, we see that a = 100,000 and b = 0.97.

General rule for modeling exponential decay

Exponential decay can be modeled with the function

y = ab^x for a > 0 and 0 < b < 1

y = ab^x

x is the exponent

a is the starting amount when x = 0

b is the base, rate, or decay factor and it is a constant and it is smaller than 1.

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