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.

For example suppose that the population of a city was 100,000 in 1980. Then every year, the population has decreased by 3% of 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(0.97)

                                                                           = 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)(0.97)

                                                                     = 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

General rule for modeling exponential decay

Exponential decay can be modeled with the function

y = abx for a > 0 and 0 < b < 1

y = abx

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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