Solving Exponential Functions: Finding the Original Amount

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. This article focuses on how to find the amount at the beginning of the time period, a.

Exponential Growth

Exponential growth: the change that occurs when an original amount is increased by a consistent rate over a period of time

Exponential Growth in Real Life:

• Values of home prices
• Values of investments
• Increased membership of a popular social networking site

Here's an exponential growth function:

y = a(1 + b)^x

• y: Final amount remaining over a period of time
• a: The original amount
• x: Time
• The growth factor is (1 + b).
• The variable, b, is percent change in decimal form.

Exponential Decay

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

Exponential Decay in Real Life:

• Decline of Newspaper Readership
• Decline of strokes in the U.S.
• Number of people remaining in a hurricane-stricken city

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 percent decrease in decimal form.

Purpose of Finding the Original Amount

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 an investment with an 8% growth rate that 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.

How to Solve for the Original Amount of an Exponential Function

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

Hint: 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: 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

Be careful: 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)

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, or the amount that your family should 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)

Practice Exercises: Answers and Explanations

Here are examples of how to solve for the original amount, given the exponential function:

1. 84 = a(1+.31)⁷
   Use Order of Operations to simplify.
   84 = a(1.31)⁷ (Parenthesis)
   84 = a(6.620626219) (Exponent)
   Divide to solve.
   84/6.620626219 = a(6.620626219)/6.620626219
   12.68762157 = 1a
   12.68762157 = a
   Use Order of Operations to check your answer.
   84 = 12.68762157(1.31)⁷ (Parenthesis)
   84 = 12.68762157(6.620626219) (Exponent)
   84 = 84 (Multiplication)
2. a(1 -.65)³ = 56
   Use Order of Operations to simplify.
   a(.35)³ = 56 (Parenthesis)
   a(.042875) = 56 (Exponent)
   Divide to solve.
   a(.042875)/.042875 = 56/.042875
   a = 1,306.122449
   Use Order of Operations to check your answer.
   a(1 -.65)³ = 56
   1,306.122449(.35)³ = 56 (Parenthesis)
   1,306.122449(.042875) = 56 (Exponent)
   56 = 56 (Multiply)
3. a(1 + .10)⁵ = 100,000
   Use Order of Operations to simplify.
   a(1.10)⁵ = 100,000 (Parenthesis)
   a(1.61051) = 100,000 (Exponent)
   Divide to solve.
   a(1.61051)/1.61051 = 100,000/1.61051
   a = 62,092.13231
   Use Order of Operations to check your answer.
   62,092.13231(1 + .10)⁵ = 100,000
   62,092.13231(1.10)⁵ = 100,000 (Parenthesis)
   62,092.13231(1.61051) = 100,000 (Exponent)
   100,000 = 100,000 (Multiply)
4. 8,200 = a(1.20)¹⁵
   Use Order of Operations to simplify.
   8,200 = a(1.20)¹⁵ (Exponent)
   8,200 = a(15.40702157)
   Divide to solve.
   8,200/15.40702157 = a(15.40702157)/15.40702157
   532.2248665 = 1a
   532.2248665 = a
   Use Order of Operations to check your answer.
   8,200 = 532.2248665(1.20)¹⁵
   8,200 = 532.2248665(15.40702157) (Exponent)
   8,200 = 8200 (Well, 8,199.9999...Just a bit of a rounding error.) (Multiply.)
5. a(1 -.33)² = 1,000
   Use Order of Operations to simplify.
   a(.67)² = 1,000 (Parenthesis)
   a(.4489) = 1,000 (Exponent)
   Divide to solve.
   a(.4489)/.4489 = 1,000/.4489
   1a = 2,227.667632
   a = 2,227.667632
   Use Order of Operations to check your answer.
   2,227.667632(1 -.33)² = 1,000
   2,227.667632(.67)² = 1,000 (Parenthesis)
   2,227.667632(.4489) = 1,000 (Exponent)
   1,000 = 1,000 (Multiply)
6. a(.25)⁴ = 750
   Use Order of Operations to simplify.
   a(.00390625)= 750 (Exponent)
   Divide to solve.
   a(.00390625)/00390625= 750/.00390625
   1a = 192,000
   a = 192,000
   Use Order of Operations to check your answer.
   192,000(.25)⁴ = 750
   192,000(.00390625) = 750
   750 = 750