# 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:

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
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
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
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
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
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