College algebra

CAmodule8lecture.docx

Module 8: Logarithmic Expressions, Equations, and Functions

Logarithm

For x > 0 and a > 0, a ≠ 1

y = logax means x=ay

The expression y = logax is read “the logarithm of x to the base a ” or simply “log, base a , of x”.

The expression y = logax represents the exponent to which the base a must be raised to obtain x.

A logarithm is an exponent.

Examples:

1) x = 2y (exponential form) is an equivalent to y = log2x (logarithmic form)

2) 34 = 81 🡪 log381=4

Graph Logarithmic Functions

For any real number a > 0, a ≠ 1, and x > 0

f(x) = logax or y= logax, is a logarithmic function

Logarithmic functions can be graphed by converting the logarithmic equation to an exponential equation and then plotting points.

For all logarithmic functions of the form:

f(x) = logax or y= logax, where a > 0, a ≠ 1, and x > 0

1. The domain of the function is (0, ∞)

2. The range of the function is (-∞, ∞)

The graph of the function passes through the points: (1a, -1), (1, 0), and (a, 1)

Example:

Graph f(x) = log2x

Logarithmic Function - Any function in the form of y= logax which is the exponent y such that ay=x.

The number a is called the base of the logarithm and a can be any positive constant other than 1.

Example: Graph the following logarithmic function by using a table to find at least three ordered pairs.

f(x) = log2x

Solution

a) Remember that y = f(x) and in this case 2𝑦=x

b) Let y = 0, 1, and 2 and plug into the function to solve for x

A ) x = 20 = 𝟏 B) x = 21 = 𝟐 C) x = 22 = 4

x	y
1	0
2	1
4	2

Source: GraphingExponentialLogarithmicFunctions.pdf (fl.edu)Links to an external site.

Properties of Logarithm

Argument: in the logarithm expression logax, x is called the argument of the logarithm.

Example:

For the logarithmic expression: log103, the argument is 3

Product Rule for Logarithms:

For positive real numbers x, y and a, a ≠ 1

logaxy= logax+ logay

Example: log3(6.7)= log36+ log37

Quotient Rule for Logarithms:

For positive real numbers x, y and a, a ≠ 1

logaxy= logax - logay

Example: log3194= log319 - log34

Power Rule for Logarithms:

If x and a are positive real numbers, a ≠ 1, and n is any real number, then: logaxn=nlogax

Example: log243=3log24

Additional Properties of Logarithms:

If a > 0, and a ≠ 1, then

logaan=x

alogax= x (x > 0)

Examples:

log665=5

Common Logarithms:

A Common Logarithms is a logarithm with a base of 10. When the base of a logarithm is not indicated; we assume the base is 10.

log x = log10x(x > 0) and all the properties studied before apply to these common logarithms.

Solve Applications of Logarithmic Functions

Logarithms are used to measure the magnitude of earthquakes. Charles R. Richter developed the Richter scale for measuring earthquakes. The magnitude, R, of an earthquake on the Richter scale is given by the formula: R= log10I

I represent the number of times greater (or more intense) the earthquake is than the smallest measurable activity that can be measured on a seismograph.

a. If an earthquake measures 4 on the Richter scale, how much more intense is it than the smallest measurable activity?

4= log10I

104 = I 🡪 10,000 = I

An earthquake that measures 4 on the Richter scale is 10,000 times more intense than the smallest measurable activity.

b. How many times more intense is an earthquake that measures 5 on the Richter scale than an earthquake that measures 4?

5= log10I

105 = I 🡪 100,000 = I

Since (10,000) (10) = 100,000, an earthquake measuring 5 on the Richter scale is 10 times more intense than an earthquake measuring 4 on the Richter scale.

References

Angel, A., & Rundle, D. (2018). Intermediate algebra for college students (10th edition). Pearson.

CAmodule8discussion.docx

Module 8: Discussion Logarithmic Expressions, Equations, and Functions

Module 8: Logarithmic Expressions, Equations, and Functions

Discussion Instructions:

Please review the class discussion directions . They tell you how to prepare and reflect before responding to the prompt below. They also include detailed guidelines how you must respond to classmates and then reply to the responses you receive. Please pay close attention to the response directions. To earn your discussion points, you must post, respond, and reply as directed.

Discussion writing prompt

Please create a new forum thread and address this prompt.

1. Explain the following statement: “A logarithm is an exponent." Provide at least two examples to support your answer.

Response focus

1. In your response, in addition to the standard requirement of noting what you have in common and where you differ with classmates, offer a suggestion as to how to improve your classmates' comfort level with the topics we are studying this week.

Remember to review the academic expectations for your submission.

Submission reminders:

· Submit your initial discussion post by 11:59 pm ET on Wednesday. Respond to two of your classmates' discussion posts, and reply to the responses you received by 11:59 pm ET on Saturday.

· Contribute a minimum of 250 words to the initial post.

· Follow established netiquette (Links to an external site.) guidelines when participating in forums.

Reply

CAmodule8reading.docx

Module 8: Essential Material

Lecture

Angel, A., & Runde, D. (2018). Intermediate algebra for college students (10th ed.). Pearson.

Chapter 9: Exponential and Logarithmic Functions Section - 9.3: Logarithmic Functions

Download and use the Word handout A Brief Guide to Logarithmic Expressions, Equations, and Functions Download A Brief Guide to Logarithmic Expressions, Equations, and Functions

Logarithmic function reference. (2017). Math is Fun. https://www.mathsisfun.com/sets/function-logarithmic.htmlLinks to an external site.

Multimedia

Plotting points of logarithmic function | Logarithms | Algebra II | Khan Academy

Khan Academy. (n.d.). Plotting points of logarithmic function | Logarithms | Algebra II | Khan Academy [Video]. YouTube. https://www.youtube.com/watch?v=DhW9pz5Vfwo

Logarithms | Logarithms | Algebra II | Khan Academy

Khan Academy. (n.d.). Logarithms | algebra II | khan academy [Video]. YouTube. https://www.youtube.com/watch?v=Z5myJ8dg_rM

Solving logarithmic equations | Exponential and logarithmic functions | Algebra II | Khan Academy

Khan Academy. (n.d.). Solving logarithmic equations [Video]. YouTube. https://www.youtube.com/watch?v=Kv2iHde7Xgw

Graphing logarithmic functions | Exponential and logarithmic functions | Algebra II | Khan Academy

Khan Academy. (n.d.). Graphing logarithmic functions | Exponential and logarithmic functions | Algebra II | Khan [Video]. YouTube. Academy https://www.youtube.com/watch?v=DuYgVVU_BwY

Solving exponential equation | Exponential and logarithmic functions | Algebra II | Khan Academy

Khan Academy. (n.d.). Solving exponential equation | Exponential and logarithmic functions | Algebra II | Khan Academy [Video]. YouTube. https://www.youtube.com/watch?v=7Ig6kVZaWoU

4.5 Applications of Exponential & Logarithmic Functions.

Spectrum Math. (n.d.). 4.5 Applications of Exponential & Logarithmic Functions. [Video]. YouTube. https://www.youtube.com/watch?v=hPvJ4h0frpo

CAmodule8lecture.docx

Module 8: Logarithmic Expressions, Equations, and Functions

Logarithm

For x > 0 and a > 0, a ≠ 1

y = logax means x=ay

The expression y = logax is read “the logarithm of x to the base a ” or simply “log, base a , of x”.

The expression y = logax represents the exponent to which the base a must be raised to obtain x.

A logarithm is an exponent.

Examples:

1) x = 2y (exponential form) is an equivalent to y = log2x (logarithmic form)

2) 34 = 81 🡪 log381=4

Graph Logarithmic Functions

For any real number a > 0, a ≠ 1, and x > 0

f(x) = logax or y= logax, is a logarithmic function

Logarithmic functions can be graphed by converting the logarithmic equation to an exponential equation and then plotting points.

For all logarithmic functions of the form:

f(x) = logax or y= logax, where a > 0, a ≠ 1, and x > 0

1. The domain of the function is (0, ∞)

2. The range of the function is (-∞, ∞)

The graph of the function passes through the points: (1a, -1), (1, 0), and (a, 1)

Example:

Graph f(x) = log2x

Logarithmic Function - Any function in the form of y= logax which is the exponent y such that ay=x.

The number a is called the base of the logarithm and a can be any positive constant other than 1.

Example: Graph the following logarithmic function by using a table to find at least three ordered pairs.

f(x) = log2x

Solution

a) Remember that y = f(x) and in this case 2𝑦=x

b) Let y = 0, 1, and 2 and plug into the function to solve for x

A ) x = 20 = 𝟏 B) x = 21 = 𝟐 C) x = 22 = 4

x	y
1	0
2	1
4	2