Hi Everyone!

Lesson 36: Properties of Logarithms & Compound Interest

### 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.

Learning Outcomes.

• Simplify a logarithmic expression.
• Expand a logarithmic expression using the properties.
• Rewrite a logarithmic expression as a single logarithm using the properties.
• Use the change-of-base formula.
• Solve compound interest problems.

Topic. This lesson covers

Section 8.5: Properties of Logarithms, and

Section 8.6: The Irrational Number $e$ and Change of Base.

WeBWorK. There are two WeBWorK assignments on today’s material:

LogarithmicProperties

CompoundInterest

Lesson Notes.

These notes are used in Lessons 34-36. Today’s lesson starts on Properties on page 3 through page 5.

These notes are used in Lessons 36 and 37. Today’s lesson starts on Application on page 2 through page 4, not including the Example.

Video Lessons.

Video Lesson 36 – part 1 (based on Lesson 36 Notes -part 1)

This video is used in Lessons 34-36. For today’s lesson, watch from [11:39] to [22:24].

Video Lesson 36 – part 2 (based on Lesson 36 Notes – part 2)

This video is used in Lessons 36 and 37. For today’s lesson, watch from [5:23] to [9:16].

### 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.

#### Warmup Question 1

Find $\log_7\sqrt7.$

$7^{1/3}=\sqrt7$, so $\log_7(\sqrt7)=1/3$.

#### Warmup Question 2

Find $\log_88.$

$8^1=8$, so $\log_88=1$.

#### Warmup Question 3

Find $\log_{15}1.$

$15^{0}= 1$, so $\log_{15}1=0$.

#### Warmup Question 4

Find $\log_aa^3.$

$\log_aa^3=3$

### 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 35

### Quick Intro I

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 Properties of Logarithms

Key Words. Logarithmic expression, properties of logarithms, change-of-base formula.

In the Warmup Questions 2-4, we saw that

$$\log_88=1,$$

$$\log_{15}1=0,$$

$$\log_aa^3=3.$$

These logarithms can be generalized. Say that $b>0$ and $b\neq 1$. Think about what the following logarithms are. Do you see the pattern with the above logarithms?

$$\log_bb$$

$$\log_{b}1$$

$$\log_bb^p$$

Below we give the answers, and provide a few more useful properties.

Properties of Logarithms

Suppose $a, b,x,y>0$ and $a, b\neq 1$.

• $\log_bb=1$
• $\log_{b}1=0$
• $\log_bb^p=p$
• $b^{\log_bx}=x$
• $\log_{b}(xy) = \log_bx+\log_by$
• $\log_{b}\left(\dfrac{x}{y}\right) = \log_bx-\log_by$
• $\log_bx^p=p\log_bx$
• Change-of-Base Formula: $\log_bx=\dfrac{\log_ax}{\log_ab}$

These properties are used to simplify a logarithmic expression as you will see in the video. They are also used to solve logarithmic equations as we will see in Lesson 37.

### Video Lesson I

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 1

A description of the video

This video is used in Lessons 34-36. For today’s lesson, watch from [11:39] to [22:24].

In this video you will see the logarithmic expressions:

• $\log_57$
• $\log_b\sqrt{xy}z^2$
• $\log_{7}\left(\dfrac{x^2y}{zw^2}\right)$
• $\log_2x-2\log_2y+\dfrac{1}{2}\left(\log_2z-\log_2w)-1$

### Try Questions I

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

Suppose $x,y>0$. Expand $\log\left(\dfrac{\sqrt{10x}}{y^2}\right)$.

$$\log\left(\dfrac{\sqrt{10x}}{y^2}\right)= \log\left(\sqrt{10x}\right)-\log y^2$$

$$=\log\left(10x\right)^{1/2}-\log y^2$$

$$= \dfrac{1}{2}\log\left(10x\right)-2\log y$$

$$=\dfrac{1}{2}\left(\log 10+\log x\right)-2\log y$$

$$=\dfrac{1}{2}\left(1+\log x\right)-2\log y$$

$$=\dfrac{1}{2}+\dfrac{\log x}{2}-2\log y$$

#### Try Question 2

Rewrite the expression as a single logarithm and simplify.

$$\log_7{x}-3\log_7y+6\log_7z$$

$$\log_7{x}-3\log_7y+6\log_7z = \log_7x- \log_7y^3+\log_7z^6$$

$$= \log_7x+\log_7z^6- \log_7y^3$$

$$= \log_7(xz^6)-\log_7y^3$$

$$=\log_7\left(\dfrac{xz^6}{y^3}\right)$$

#### Try Question 3

Calculate $\log_{100}5$. Round your answer to the nearest thousandth.

$$\log_{100}5 = \dfrac{\log 5}{\log 100} = \dfrac{0.69897}{2}\approx 0.349$$

### WeBWorK I

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 I

It is time to do the homework on WeBWork:

LogarithmicProperties

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

### Exit Questions I

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

• In $3^x=4$, does the base of the logarithm used when solve that equation matter?
• How do the properties of logarithms correspond to properties of exponents?

$\bigstar$  (a) Assuming that $x,y,z>0$, combine to an expression with one logarithm only.

$$\dfrac{2}{3}\log_5 x – 5\log_5 y-3\log_5z$$

$\bigstar$ (b) Assuming that $x,y>0$, expand $$\log_6\left(\dfrac{\sqrt{x^2}}{36y^3}\right) .$$

(a) $$\dfrac{2}{3}\log_5 x – 5\log_5 y-3\log_5z=\log_5 x^{2/3}-\log_5y^5-\log_5z^3$$

$$= \log_5\sqrt{x^2}-(\log_5y^5+\log_5z^3)$$

$$= \log_5\sqrt{x^2}-\log_5(y^5z^3)$$

$$= \log_5\left(\dfrac{\sqrt{x^2}}{y^5z^3}\right)$$

(b) $$\log_6\left(\dfrac{\sqrt{x^2}}{36y^3}\right) = \log_6\sqrt{x^2}-\log_6(36y^3)$$

$$=\dfrac{2}{5}\log_6x-(\log_6{36}+\log_6y^3)$$

$$= \dfrac{2}{5}\log_6x-(2+\log_6y^3)$$

$$=\dfrac{2}{5}\log_6x-\log_6y^3-2$$