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Lesson 26: Logarithmic Expressions

### Resources

In this section you will find some important information about the specific resources related to this lesson:

• the learning outcomes,
• the section in the textbook,
• the WeBWorK homework sets,
• a link to the pdf of the lesson notes,
• a link to a video lesson.

Resources

Learning Outcomes.

• Evaluate a logarithmic expression.
• Write a logarithmic expression in exponential form.
• Write an exponential expression in logarithmic form.

Topic. This lesson covers

Section 8.4: Logarithmic Functions.

WeBWorK. There is one WeBWorK assignment on today’s material:

LogarithmicFunctions

Lesson Notes.

These notes are used in Lessons 25-27. Today’s lesson is based on the examples, pages 2-3.

Video Lesson.

Video Lesson 26 (based on Lesson 26 Notes)

This video is used in Lessons 25-27. For today’s lesson, watch from [4:37] to [11:38].

### Warmup Questions

These are questions on fundamental concepts that you need to know before you can embark on this lesson. Don’t skip them! Take your time to do them, and check your answer by clicking on the “Show Answer” tab.

Solve

, so .

Solve

, so .

Solve

, so .

Solve

, so .

### Review

If you are not comfortable with the Warmup Questions, don’t give up! Click on the indicated lesson for a quick catchup. A brief review will help you boost your confidence to start the new lesson, and that’s perfectly fine.

Need a review? Check Lesson 1, Lesson 4, and Lesson 25.

### Quick Intro

This is like a mini-lesson with an overview of the main objects of study. It will often contain a list of key words, definitions and properties – all that is new in this lesson. We will use this opportunity to make connections with other concepts. It can be also used as a review of the lesson.

A Quick Intro to Logarithmic Expressions

Key Words. Logarithmic expression, logarithm, base, argument, exponential form, logarithmic form, natural logarithm, common logarithm.

In the Warmup Question 2 we saw when . Another way of describing this is to say that is the logarithm of 81 with base , and write:

In general, for positive real numbers and with , the logarithmic expression

means that

In the expression :

• is the logarithm,
• is the base, and
• is the argument.

The above definition relates a logarithmic form to an exponential form (recall Lesson 25 on Exponential Expressions). For example:

The logarithmic form of is .

The exponential form of is .

Some bases are special.

When is the base, instead of writing , we write . This logarithm is called the natural logarithm.

When is the base, instead of writing , we write . This logarithm is called the common logarithm.

### Video Lesson

Many times the mini-lesson will not be enough for you to start working on the problems. You need to see someone explaining the material to you. In the video you will find a variety of examples, solved step-by-step – starting from a simple one to a more complex one. Feel free to play them as many times as you need. Pause, rewind, replay, stop… follow your pace!

Video Lesson

This video is used in Lessons 25-27. For today’s lesson, watch from [4:37] to [11:38].

A description of the video

In this video you will see the logarithmic expressions:

### Try Questions

Now that you have read the material and watched the video, it is your turn to put in practice what you have learned. We encourage you to try the Try Questions on your own. When you are done, click on the “Show answer” tab to see if you got the correct answer.

#### Try Question 1

Write in exponential form.

#### Try Question 2

Write in logarithmic form.

Evaluate .

, so .

Evaluate

, so .

### WeBWorK

You should now be ready to start working on the WeBWorK problems. Doing the homework is an essential part of learning. It will help you practice the lesson and reinforce your knowledge.

WeBWorK

It is time to do the homework on WeBWork:

LogarithmicFunctions

When you are done, come back to this page for the Exit Questions.

### Exit Questions

After doing the WeBWorK problems, come back to this page. The Exit Questions include vocabulary checking and conceptual questions. Knowing the vocabulary accurately is important for us to communicate. You will also find one last problem. All these questions will give you an idea as to whether or not you have mastered the material. Remember: the “Show Answer” tab is there for you to check your work!

#### Exit Questions

• How is approximating the solution to similar to solving on a calculator?
• What is ?
• In the definition of , why don’t we allow to be ?

Evaluate