How to Solve Exponential Decay Functions

Exponential functions tell the stories of explosive change. The two types of exponential functions are exponential growth and exponential decay. Four variables (percent change, time, the amount at the beginning of the time period, and the amount at the end of the time period) play roles in exponential functions. Use an exponential decay function to find the amount at the beginning of the time period.

Exponential Decay

Exponential decay is the change that occurs when an original amount is reduced by a consistent rate over a period of time.

Here's an exponential decay function:

y = a(1-b)^x

• y: Final amount remaining after the decay over a period of time
• a: The original amount
• x: Time
• The decay factor is (1-b)
• The variable b is the percent of the decrease in decimal form.

Purpose of Finding the Original Amount

If you are reading this article, then you are probably ambitious. Six years from now, perhaps you want to pursue an undergraduate degree at Dream University. With a $120,000 price tag, Dream University evokes financial night terrors. After sleepless nights, you, Mom, and Dad meet with a financial planner. Your parents' bloodshot eyes clear up when the planner reveals that an investment with an eight percent growth rate can help your family reach the $120,000 target. Study hard. If you and your parents invest $75,620.36 today, then Dream University will become your reality thanks to exponential decay.

How to Solve

This function describes the exponential growth of the investment:

120,000 = a(1 +.08)⁶

• 120,000: Final amount remaining after 6 years
• .08: Yearly growth rate
• 6: The number of years for the investment to grow
• a: The initial amount that your family invested

Thanks to the symmetric property of equality, 120,000 = a(1 +.08)⁶ is the same as a(1 +.08)⁶ = 120,000. Symmetric property of equality states that if 10 + 5 = 15, then 15 = 10 + 5.

If you prefer to rewrite the equation with the constant (120,000) on the right of the equation, then do so.

a(1 +.08)⁶ = 120,000

Granted, the equation doesn't look like a linear equation (6a = $120,000), but it's solvable. Stick with it!

a(1 +.08)⁶ = 120,000

Do not solve this exponential equation by dividing 120,000 by 6. It's a tempting math no-no.

1. Use order of operations to simplify

a(1 +.08)⁶ = 120,000
a(1.08)⁶ = 120,000 (Parenthesis)
a(1.586874323) = 120,000 (Exponent)

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2. Solve by dividing

a(1.586874323) = 120,000
a(1.586874323) / (1.586874323) = 120,000 / (1.586874323)
1a = 75,620.35523
a = 75,620.35523

The original amount to invest is approximately $75,620.36.

3. Freeze: You're not done yet; use order of operations to check your answer

120,000 = a(1 +.08)⁶
120,000 = 75,620.35523(1 +.08)⁶
120,000 = 75,620.35523(1.08)⁶ (Parenthesis)
120,000 = 75,620.35523(1.586874323) (Exponent)
120,000 = 120,000 (Multiplication)

Answers and Explanations to the Questions

Woodforest, Texas, a suburb of Houston, is determined to close the digital divide in its community. A few years ago, community leaders discovered that their citizens were computer illiterate. They did not have access to the internet and were shut out of the information superhighway. The leaders established the World Wide Web on Wheels, a set of mobile computer stations.

World Wide Web on Wheels has achieved its goal of only 100 computer illiterate citizens in Woodforest. Community leaders studied the monthly progress of the World Wide Web on Wheels. According to the data, the decline of computer illiterate citizens can be described by the following function:

100 = a(1 - .12)¹⁰

1. How many people are computer illiterate 10 months after the inception of the World Wide Web on Wheels?

• 100 people

Compare this function to the original exponential growth function:

100 = a(1 - .12)¹⁰
y = a(1 + b)^x

The variable y represents the number of computer illiterate people at the end of 10 months, so 100 people are still computer illiterate after the World Wide Web on Wheels began to work in the community.

2. Does this function represent exponential decay or exponential growth?

• This function represents exponential decay because a negative sign sits in front of the percent change (.12).

3. What is the monthly rate of change?

• 12 percent

4. How many people were computer illiterate 10 months ago, at the inception of the World Wide Web on Wheels?

• 359 people

Use ​order of operations to simplify.

100 = a(1 - .12)¹⁰

100 = a(.88)¹⁰ (Parenthesis)

100 = a(.278500976) (Exponent)

Divide to solve.

100(.278500976) = a(.278500976) / (.278500976)

359.0651689 = 1a

359.0651689 = a

Use the order of operations to check your answer.

100 = 359.0651689(1 - .12)¹⁰

100 = 359.0651689(.88)¹⁰ (Parenthesis)

100 = 359.0651689(.278500976) (Exponent)

100 = 100 (Multiply)

5. If these trends continue, how many people will be computer illiterate 15 months after the inception of the World Wide Web on Wheels?

• 52 people

Add in what you know about the function.

y = 359.0651689(1 - .12) ^x

y = 359.0651689(1 - .12) ¹⁵

Use Order of Operations to find y.

y = 359.0651689(.88)¹⁵ (Parenthesis)

y = 359.0651689 (.146973854) (Exponent)

y = 52.77319167 (Multiply).