Video Transcript: Algebra Nspirations: Exponents and Exponential Functions, Part 2
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Algebra >> Exponential and Logarithmic Functions >> Applications of Exponential and Logarithmic Functions

Video Transcript: Algebra Nspirations: Exponents, 2

Video Transcript: Algebra Nspirations: Exponents and Exponential Functions, Part 2

This is the transcript for the video of same title. Video contents: In this Investigation we look at exponential growth and decay models.

Exponential Functions

Brief Review of Exponents

When a number is raised to a power, exponents are involved. Here are the key components of an exponential expression.

Exponents make it easier to write expressions that would otherwise be cumbersome to write. See how much easier it is to write 310 instead of the expression on the right? 

3 to the power of 10 equals 3 times 3 times 3 times 3 times 3 times 3 times 3 times 3 times 3 times 3

Exponents are a different form of writing numbers. With this new form come new properties that are unique to exponents. The next section goes over these properties, also known as the Laws of Exponents.

Laws of Exponents

The Laws of Exponents are a set of properties that pertain to all exponential expressions. Study the definition below.

Laws of Exponents. The rules for simplifying exponents when multiplying or dividing expressions with a common base and different exponents.

To learn more about exponents, click on this link. It is a presentation that goes over the Laws of Exponents in more detail. To see examples that use the Laws of Exponents, click on this link. 

 

Exponential Expressions

Example 1

What is the value of x?

8 to the power of x equals 32

In order to work with the Laws of Exponents, make sure the terms have the same base. Both 8 and 32 can be written as powers of 2:

8 equals 2 cubed and 32 equals 2 to the power of 5

Rewrite the original expression with these powers of 2:

2 to the power of 3 x equals 2 to the power of 5

With a common base, you can now solve for x:

Solving the equation 3 x = 5.

Example 2

If 4x - 5y = 10, what is the value of this expression?

16 to the power of x over 2 to the power of y

Find a common base for the numerator and denominator, then simplify the expression:

2 to the power of 4 x over 2 to the power of 5 y equals 2 to the power of 4 x minus 5 y

Do you notice that the exponent for 2 is the same as the left side of the linear equation? Replace the exponents

Solving an exponential equation.

Example 3

Rewrite this expression as an exponential expression with a power of 5.

fourth root of 125

First, write the numerical expression as an exponential expression:

fourth root of 125 equals 125 to the power of 1 fourth

Next, rewrite 125 as a power of 5:

125 to the power of 1 fourth equals open parentheses 5 cubed close parentheses to the power of 1 fourth

Finally, rewrite the entire expression:

fourth root of 125 equals 5 to the power of 3 over 4

Exponential Functions

An exponential function is of this form:

{"mathml":"<math style=\"font-family:stix;font-size:36px;\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mstyle mathsize=\"36px\"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mi>a</mi><mo>&#x2022;</mo><msup><mi>b</mi><mrow><mi>c</mi><mi>x</mi></mrow></msup></mstyle></math>","truncated":false}

In this function the terms a, b, and c are numbers and x is the independent variable. Here’s one of the simplest exponential functions:

{"mathml":"<math style=\"font-family:stix;font-size:36px;\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mstyle mathsize=\"36px\"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><msup><mn>2</mn><mi>x</mi></msup></mstyle></math>","truncated":false}

This is the graph of the function.

Graph of an exponential function.

As you can see from the graph, the values of f(x) increase dramatically for small increases in input values of x. Here are some other examples of exponential functions.

 

Graph of an exponential function.

Graph of an exponential function.

Graph of an exponential function.

Equation for an exponential function

Equation for an exponential function

Equation for an exponential function

 

Compound Interest

A special case involving exponential expressions is compound interest. What is compound interest? Usually, the context is of money earning interest in a bank account.

Here is the compound interest formula

The compound interest formula for the case where the compounding is non-continuous.

This is the formula to use when the compounding periods are non-continuous. Use this formula for continuous compounding:

The compound interest formula for the case where the compounding is continuous.

To see examples of calculating compound interest, click on this link. 

Compound Interest

Example 1

A savings account earns interest according the formula below. What is the exponential function that models this growth? How much is in the savings account after five years?

“Each year the amount of the account increases by 2.5%. The initial investment was $1750.”

Use the compound interest placeholder formula:

A equals P open parentheses 1 plus r over n close parentheses to the power of n t

In this formula:

A = Current amount in savings

P = Initial investment

r = Interest rate

n = Number of compounding periods

t = Number of years

We know the following:

P = 1750

r = 0.025

n = 1

We can now generate the exponential function that models the growth:

A of T equals 1750 times 1.025 to the power of t

We can use this model to find the amount in savings after 5 years:

Evaluating A of T equals 1750 times 1.025 to the power of t, for t = 5.

This is part of a collection of video transcript from the Algebra Nspirations video series. To see the complete collection of transcripts, click on this link.

Note: The download is a PDF file.

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Common Core Standards CCSS.Math.CONTENT.HSF.LE.A.2
Grade Range 6 - 12
Curriculum Nodes Algebra
    • Exponential and Logarithmic Functions
        • Applications of Exponential and Logarithmic Functions
        • Graphs of Exponential and Logarithmic Functions
Copyright Year 2011
Keywords exponents, graphs of exponential functions, exponential equations, algebra resources for teachers, algebra resources for teachers and students, video transcript