Name: A2.L8-4.WU Date: Period:

THINK, PAIR, SHARE

Introduction to Logarithms

1. Look at the slip of paper you have received. 2. For each equation in exponential form, label the 𝒃𝒂𝒔𝒆 (𝑩), 𝒆𝒙𝒑𝒐𝒏𝒆𝒏𝒕 (𝒀) and 𝒑𝒓𝒐𝒅𝒖𝒄𝒕(𝑿). (If you’re paper has logarithmic form do not do this step.) 3. Find the person in the room who has a slip of paper with the same numbers as you. (Their numbers can be physically smaller or be in a different position from yours, but the value must be the same.) 4. Assuming the base, exponent, and product have not changed, label each in the logarithmic equation on your partner’s slip of paper. 5. Use the words base, exponent, and product to explain what you think a “log” is or does.

WARM-UP

Exponent Properties

Use your knowledge of exponent properties to answer the following questions.

𝑎1

2 = √𝑎2 𝑎1

3 = √𝑎3 𝑎−𝑛 =1

𝑎𝑛

1. Which of the following is the expression below in simplest form?

(𝑝2𝑞−3)−1

A. 𝑝−2𝑞3 B. 𝑝2𝑞3

C. 𝑝2

𝑞3

D. 𝑞3

𝑝2

2. Which of the following is equivalent to the expression below?

6412

A. 4 B. 8 C. 32 D. 128

A2.L8-4.Notes

LESSON 8-4: LOGARITHMIC FUNCTIONS AS INVERSES Algebra Objective Students will be able to graph logarithmic functions and evaluate logarithmic

expressions. Language Objective Students will describe the process of evaluating a logarithm and justify their

choice to express a number as a particular power of a common base.

Big Idea

An exponential function of the form 𝑦 = 𝑏𝑥 has an inverse of the form 𝑥 = 𝑏𝑦 . To express “𝑦 𝑎𝑠 𝑎 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 𝑜𝑓 𝑥” for the inverse, we write 𝑦 = log𝑏 𝑥.

You can read log𝑏 𝑥 as “𝑙𝑜𝑔 𝑏𝑎𝑠𝑒 𝑏 𝑜𝑓 𝑥.” In other words, the logarithm y is the exponent to which

be must be raised to get x.

𝒚 = 𝒍𝒐𝒈𝒃𝒙 (Logarithmic Form) 𝒙 = 𝒃𝒚 (Exponential Form)

If the logarithm does not have a base, we assume that the base is _______________.

Directions: Draw a line from each logarithm equation to its exponential equation in Column B.

Column A Column B 1. 𝑙𝑜𝑔2 16 = 4 A. 103 = 1000

2. 𝑙𝑜𝑔3 9 = 2 B. 𝑏𝑦 = 𝑥

3. 𝑙𝑜𝑔 1000 = 3 C. 32 = 9

4. 𝑙𝑜𝑔𝑏 𝑥 = 𝑦 D. 24 = 16

Write each equation in exponential form. Use mental math to solve each equation.

5. 𝑦 = 𝑙𝑜𝑔3 27

6. 𝑦 = 𝑙𝑜𝑔5 25

A2.L8-4.Notes

Example #1

Evaluate 𝑙𝑜𝑔2 64

𝑙𝑜𝑔2 64 = 𝑥

First, write in exponential form.

2𝑥 = 64

Then, find a common base.

2𝑥 = 26

Solve for the missing exponent.

2𝑥 = 26

𝑥 = 6

∴ 𝑙𝑜𝑔2 64 = 𝟔

Check:

26 = 2 ∙ 2 ∙ 2 ∙ 2 ∙ 2 ∙ 2 = 64

Example #2

Evaluate 𝑙𝑜𝑔3 81

Example #3

Evaluate 𝑙𝑜𝑔4 32

Example #4

Evaluate 𝑙𝑜𝑔8 16

Example #5

Evaluate 𝑙𝑜𝑔10 0.01

Example #6

Evaluate 𝑙𝑜𝑔2 0.5

A2.L8-4.Notes

Investigation 10-4: Graphs of Logarithmic Functions

Directions: Fill in the table below using your knowledge of logarithms and exponential functions. Then plot the points on the graph provided. Do you notice any symmetry between the graphs?

𝑦 = 𝑙𝑜𝑔2𝑥

𝑦 = 2𝑥

𝑥 𝑦

𝑥 𝑦

1

8 −3

1

4 −2

1

2 −1

1 0

2 1

2

3

4

A2.L8-4.Notes

CLASSWORK 10-4

Evaluating Logarithms

Evaluate each logarithm. (Complete #1-3, and at least 2 more problems for credit.)

1. log4 64

2. log5 625

3. log11 121

4. log10 0.1

5. log31

9

6. log2 0.25

7. log48 8. log927 9. log1664

Due by the end of class.

Name: A2.L10-4.HW

HOMEWORK 10-4

Logarithms and Logarithmic Functions

Directions: Write each equation in logarithmic form.

1. 92 = 81 2.

1

64= (

1

4)

3

3. 83 = 512

4. (1

3)

−2

= 9

5. 29 = 512 6. 45 = 1024 7. 54 = 625 8. 10−3 = 0.001

Directions: Evaluate each logarithm.

9. log2 128 10. log 100,000

11. log2(−32) 12. log7 76

13. log9 27 14. log4 32

15. log31

81

16. log1

3

1

9