Hi Everyone!

On this page you will find some material about Lesson 35. Read through the material below, watch the videos, and follow up with your instructor if you have questions.

**Lesson 35: Logarithmic Expressions**

**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 34-36. Today’s lesson is based on the examples, pages 2-3.

**Video Lesson. **

Video Lesson 35 (based on Lesson 35 Notes)

This video is used in Lessons 34-36. 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.*

#### Warmup Question 1

Solve

$$3^x=3.$$

#### Show Answer 1

$3^1=3$, so $x=1$.

#### Warmup Question 2

Solve

$$3^x=81.$$

#### Show Answer 2

$3^4=81$, so $x=4$.

#### Warmup Question 3

Solve $$9^x=3.$$

#### Show Answer 3

$9^{1/2}= \sqrt{9} = 3$, so $x=1/2$.

#### Warmup Question 4

Solve $\left(\dfrac{1}{2}\right)^x=\dfrac{1}{16}.$

#### Show Answer 4

$\left(\dfrac{1}{2}\right)^{4} = \dfrac{1^4}{2^4}=\dfrac{1}{16}$, so $x=4$.

### 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

### 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 $3^x=81$ when $x=4$. Another way of describing this is to say that $4$ is the logarithm of 81 with base $3$, and write:

$$4=\log_3(81).$$

$\bigstar$ In general, for positive real numbers $x$ and $b$ with $b\neq 1$, the **logarithmic expression**

$$y=\log_bx$$

means that

$$b^y=x.$$

In the expression $y=\log_bx$:

- $y$ is the
**logarithm**, - $b$ is the
**base**, and - $x$ is the
**argument.**

$\bigstar$ The above definition relates a logarithmic form to an exponential form (recall Lesson 34 on Exponential Expressions). For example:

$\qquad\bullet$ The **logarithmic form** of $\left(\dfrac{1}{2}\right)^{4}=\dfrac{1}{16}$ is $\log_{1/2}\left(\dfrac{1}{16}\right) = 4$.

$\qquad\bullet$ The **exponential form** of $\log_93 = \dfrac{1}{2}$ is $9^{1/2}= 3$.

$\bigstar$ Some bases are special.

$\qquad\bullet$ When $b=e\approx 2.718$ is the base, instead of writing $\log_ex$, we write $\ln x$. This logarithm is called the **natural logarithm**.

$\qquad\bullet$ When $b=10$ is the base, instead of writing $\log_{10}x$, we write $\log x$. 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 34-36. 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:

- $\log_28$
- $\log_5\left(\dfrac{1}{25}\right)$
- $\log_b1$
- $\log_{10}5$
- $\ln e^2$

### 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 $\log_4 16=2$ in exponential form.

#### Show Answer 1

$4^2=16$

#### Try Question 2

Write $10^{-3}=0.001$ in logarithmic form.

#### Show Answer 2

$\log 0.001 =-3$

#### Try Question 3

Evaluate $\log_3 \left\dfrac{1}{27}$.

#### Show Answer 3

$3^{-3} = \dfrac{1}{3^3} = \dfrac{1}{27}$, so $\log_3 \left\dfrac{1}{27}=-3$.

#### Try Question 4

Evaluate $\ln \sqrt[3]e.$

#### Show Answer 4

$e^{1/3} = \sqrt[3]x$, so $\ln \sqrt[3]e=\dfrac{1}{3}$.

### 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 $10^x=2$ similar to solving $x^2=5$ on a calculator?
- What is $\log_b a$?
- In the definition of $\log_b a$, why don’t we allow $b$ to be $1$?

$\bigstar$ Evaluate $\log_{1/5}25.$

#### Show Answer

$\left(\dfrac{1}{5}\right)^{-2}=5^2=25,$ so $\log_{1/5}25=-2.$

### Need more help?

*Don’t wait too long to do the following.*

- Watch the additional video resources.

- Talk to your instructor.
- Form a study group.
- Visit a tutor. For more information, check the tutoring page.