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 Precalculus: Exponents, Roots and Logarithms

Exponents, roots, and logarithms are closely related operations with important business applications. They are commonly used by bankers to determine the present value and future value of investments. Economists also make frequent use of exponents to describe cost and production functions, as well as demand curves. The following animation uses simple examples to show the relationship between exponents, roots, and logarithms.

 ExponentsView animation
When does a banker use exponents?

Exponents

An exponent indicates that a number (the base) must be multiplied by itself a specified number of times (the exponent). Consider the following example.

In this example, the base, 2, is multiplied by itself the number of times specified by the exponent, 5. The result is 32. This mathematical notation is expressed verbally as "Two to the fifth power is 32."

Quantities that grow exponentially grow quickly. This has important implications, particularly for investments with compound interest. Consider the investments graphed below. In both cases, \$10,000 was invested for 20 years at 12 percent interest.

You can use Excel to solve problems that use exponents.
Exponents Excel Tutorial

Notice the difference between the future value of each investment. The investment with compound interest increases significantly, while the investment with simple interest grows relatively slowly. The reason for the difference can be found by looking at the formulas for the two investments.

The following formula is used to calculate the future value of an investment with compound interest. It contains an exponent, which causes the line to curve and the growth of the investment to accelerate.

Compare the compound interest formula above to the formula for simple interest below.

As you can see, the two formulas are very similar. However, because the simple interest formula does not contain an exponent, an investment with simple interest grows in a gradual, linear fashion.

If you need help with the basics of Excel, view this tutorial.
Basic Excel Tutorial

Understanding how the exponent in the compound interest formula dramatically increases the future value of the investment is critical to the field of banking, both for the banker and the client.

Negative Exponents

The preceding examples reviewed positive exponents; what happens if the exponent is negative?

A base raised to a negative power, n, is simply the reciprocal of the base raised to the positive power, n. Consider the following example.

Imagine that you would like to retire in 20 years with \$1 million in your retirement account. Assume that you can earn a 12 percent annual rate of return on this account, compounded quarterly. How much will you have to invest today as a one-time contribution to be able to retire in 20 years with \$1 million?

The amount you would have to invest today is called the present value. To solve for the present value, you would use the formula for the present value of an investment with compound interest:

PV = Present value
CN = Cash flow at time period N
r = periodic rate of interest, which is calculated as

N = number of periods, which is equal to the number of years multiplied by the number of periods per year

Using the information from the example above, identify the values for each variable:

PV= Present value
CN = \$1 million
r = (12% divided by 4 periods per year)
N = 80 (4 periods multiplied by 20 years)

Substitute the values of CN, r, and N in the formula, and compute the present value.

You would have to invest \$93,977.10 now at 12 percent compounded quarterly to have \$1,000,000 in 20 years.

Roots

Fractional exponents also can be expressed as roots.

To understand how roots operate, try solving the following equation:

First, raise both sides of the equation to the power (the reciprocal of the original exponent, 4). This will isolate the variable y.

Then, express as a root, and determine the value of y.

The fourth root of 81 is 3. In other words, 3 multiplied by itself 4 times is equal to 81.

Exponential fractions that have numbers other than 1 in the numerator can also be expressed as roots.

Try this example. What is the cube root of 8 squared? There are two approaches you could use to solve this problem. Both are shown below in parallel.

The cube root of 8 squared is 4.

As you may have guessed, fractional exponents are used in business as well. Imagine that you plan to buy a house in nine months. The down payment is \$25,000. You currently have \$20,000 invested in a certificate of deposit that compounds annually at an interest rate of 12 percent. Will your \$20,000 grow sufficiently over the nine months to cover the \$25,000 you need at the time of closing, or will you have to save additional money in the meantime?

To answer this question, use the future value formula presented below.

FV = C0 × (1 + r)N

FV = Future value
C0 = initial investment
r = periodic rate of interest, which is calculated as
N = number of periods

Using the information in the example above, determine the values for each variable or constant.

FV = Future value
C0 = \$20,000
r = (12% compounded annually)
N = (1 period for of a year)

Substitute the values of C0, r, and N in the equation, and compute the future value.

In nine months, the value of \$20,000 invested at 12 percent interest compounded annually will be \$21,774.27. You will have to save additional money prior to closing to reach your \$25,000 down payment.

 Polynomials

Logarithms

Logarithms are often used to solve mathematical expressions that include exponents. For example, how would you solve this expression for x?

One way to think about this is to ask yourself this question: "To what power must the number 10 be raised to equal 100?" In this simple example, you probably already recognize the answer: x = 2.

 Laws of Logarithms

Another way to solve this expression is by using logarithms. This method is especially useful for more complex expressions like the ones you'll encounter in your MBA coursework and in business applications. For now, reconsider the simple expression from above, and solve it using logarithms.

To solve for x using logarithms, first express this question as a logarithm and then solve for x using a logarithm table, the logarithm function on your calculator, or the logarithm function in a spreadsheet application such as Excel. The relationship 10x = 100 can be expressed using logarithm notation, as follows:

This expression is stated verbally as "log base 10 of 100 is 2." That means the base, 10, raised to the power of 2 is 100. Logarithms with a base of 10 are called common logarithms. When common logarithms are written, the base is usually omitted. For example, the logarithm shown above, x = log10100, would usually be written x = log 100.

As with exponents and roots, a common business application for logarithms is determining the present value and future value of investments. For example, you can use logarithms to determine how much time it will take an investment to reach a certain value. Imagine that you have \$1,000 invested at 8 percent compounded annually, and you want to know how long it will take for your money to double. You can use the formula for future value to solve this problem.

FV = C0 × (1 + r)N

FV = future value
C0 = initial investment
r = periodic rate of interest, which is calculated as
N = number of periods

Using the information in the example above, determine the values for each variable or constant.

FV = \$2,000 (the initial investment doubled)
C0 = \$1,000
r = (8% compounded once a year)
N = number of periods

To find the value of N, you will need to use logarithms. First, simplify the equation and divide each side by 1,000. Then take the logarithm of each side of the equation. Once you have done that, you can further simplify the equation by applying the law of logarithms that states that the logarithm of a power is the exponent times the logarithm of the base.

It will take 9.01 years for your investment to double if invested at an annual rate of 8 percent compounded annually.

You can use Excel to solve problems that use logarithms.
Logarithms Excel Tutorial

Natural logarithms

The two standard bases for logarithms are 10 and e. Logarithms with a base of 10 are called common logarithms; if a logarithm seems to be missing a base, the base of 10 is implied.

Logarithms with a base of e are called natural logarithms. These logarithms, also very common, are written using the "ln" notation instead of "log." For example, loge x is written as ln x. Natural logarithms are important primarily because they allow you to solve complicated exponential expressions that use e as the exponential base. The constant e is approximately 2.71828182845. It occurs naturally in many situations, including the countinuous compounding of interest.

In order to understand how to use the natural logarithm, consider a simple finance calculation that involves continuous compounding of interest. Suppose you have \$2,000 to invest at an annual rate of 11 percent with continuous compounding of interest. How long will it take for your initial investment to double?

To solve this problem, use the formula for future value with continuous compounding.

FV = future value
C0 = initial investment
R = annual rate of interest
T = time period in years

Using the information in the example above, determine the values for each variable or constant:

FV = \$4,000 (the initial investment doubled)
C0 = \$2,000
R = .11
T = time period in years

To find the value of T, use logarithms. First, simplify the equation and divide each side by 2,000. Then take the natural logarithm of each side of the equation. Once you have done that, you can further simplify the equation by applying the law of logarithms that states that the log of a power is the exponent times the logarithm of the base.

It will take 6.3 years for \$2,000 compounded annually at 11 percent to double.

1. Solve for x.
x = (18 + 2)2

2. Solve for x.
x = (22)4× 33

3. Solve for x.
x = 8-2

4. Simplify the expression.

5. Write as an exponential expression.

6. Write as a root.

7. What does ln e equal?

8. At an automobile company, the number of units produced, f(x), is dependent on the number of employees, x. This company's production function is f(x) = 2(x – 20)2 + 8. How many units will be produced when there are 14 employees?

9. A company's long-run production function is , where q is the number of units produced, K is the amount of capital used per day, and L is the amount of labor used per day. How many units are produced if daily capital is \$256 and daily labor is 81 individuals?

10. You would like to replace your current car in two years, and you anticipate that a new car will cost you \$25,000. If you prefer to pay for your car in cash, how much money do you need to set aside today in an investment that returns an annual rate of 12 percent compounded quarterly?

11. An individual owns 50 shares of stock; each will be worth \$19 in nine months. The value is considered to be growing at an annual rate of 11 percent compounded monthly. Determine the present value of the shares.

12. An individual invests \$55 in an account that has an interest rate of 8 percent compounded annually. After four months, what will this investment be worth?

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