Lesson 8: The “WhatPower” Function€¦ · Lesson 8: The “WhatPower” Function This file...

NYS COMMON CORE MATHEMATICS CURRICULUM M3 Lesson 8

ALGEBRA II

Lesson 8: The “WhatPower” Function

116

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Student Outcomes

Students calculate a simple logarithm using the definition.

Lesson Notes

The term logarithm is foreign and can be intimidating, so the lesson begins with a simple renaming of the logarithm

function to the more intuitive “WhatPower” function. Do not explain this function to students directly, but let them

figure out what the function does. The first two exercises have already been solved to provide a hint of how the

“WhatPower” function works.

This lesson is the first introduction to logarithms, and the work done here prepares students to solve exponential

equations of the form 𝑎𝑏𝑐𝑡 = 𝑑 (F-LE.A.4) and use logarithms to model relationships between two quantities

(F-BF.B.4a) in later lessons. In the next lessons, students create logarithm tables to discover some of the basic

properties of logarithms before continuing on to look at the graphs of logarithmic functions and then to finally modeling

logarithmic data. In this lesson, the ideas and notation of logarithmic expressions are developed, leaving many ideas to

be explored later in the module.

Classwork

Opening Exercise (12 minutes)

Allow students to work in pairs or small groups to complete these exercises. Do not explain this function to students

directly, but allow them to struggle to figure out what this new “WhatPower” function means and how to evaluate these

expressions. When there are about two minutes left, instruct groups that have not finished part (a) to skip to part (b) so

that all groups have time to think about and state the definition of this function. Consider collecting the groups’

definitions on paper and sharing some or all of them with the class using the document camera. This definition is refined

through the lessons; in particular, we are interested in the allowable values of the base 𝑏.

Opening Exercise

a. Evaluate each expression. The first two have been completed for you.

i. 𝐖𝐡𝐚𝐭𝐏𝐨𝐰𝐞𝐫𝟐(𝟖) = 𝟑

𝟑, because 𝟐𝟑 = 𝟖

ii. 𝐖𝐡𝐚𝐭𝐏𝐨𝐰𝐞𝐫𝟑(𝟗) = 𝟐

𝟐, because 𝟑𝟐 = 𝟗

iii. 𝐖𝐡𝐚𝐭𝐏𝐨𝐰𝐞𝐫𝟔(𝟑𝟔) = ______

𝟐, because 𝟔𝟐 = 𝟑𝟔

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iv. 𝐖𝐡𝐚𝐭𝐏𝐨𝐰𝐞𝐫𝟐(𝟑𝟐) = ______

𝟓, because 𝟐𝟓 = 𝟑𝟐

v. 𝐖𝐡𝐚𝐭𝐏𝐨𝐰𝐞𝐫𝟏𝟎(𝟏𝟎𝟎𝟎) = ______

𝟑, because 𝟏𝟎𝟑 = 𝟏𝟎𝟎𝟎

vi. 𝐖𝐡𝐚𝐭𝐏𝐨𝐰𝐞𝐫𝟏𝟎(𝟏𝟎𝟎𝟎𝟎𝟎𝟎) = ______

𝟔, because 𝟏𝟎𝟔 = 𝟏, 𝟎𝟎𝟎, 𝟎𝟎𝟎

vii. 𝐖𝐡𝐚𝐭𝐏𝐨𝐰𝐞𝐫𝟏𝟎𝟎(𝟏𝟎𝟎𝟎𝟎𝟎𝟎) = ______

𝟑, because 𝟏𝟎𝟎𝟑 = 𝟏, 𝟎𝟎𝟎, 𝟎𝟎𝟎

viii. 𝐖𝐡𝐚𝐭𝐏𝐨𝐰𝐞𝐫𝟒(𝟔𝟒) = ______

𝟑, because 𝟒𝟑 = 𝟔𝟒

ix. 𝐖𝐡𝐚𝐭𝐏𝐨𝐰𝐞𝐫𝟐(𝟔𝟒) = ______

𝟔, because 𝟐𝟔 = 𝟔𝟒

x. 𝐖𝐡𝐚𝐭𝐏𝐨𝐰𝐞𝐫𝟗(𝟑) = ______

𝟏

𝟐, because 𝟗

𝟏

𝟐 = 𝟑

xi. 𝐖𝐡𝐚𝐭𝐏𝐨𝐰𝐞𝐫𝟓(√𝟓) = ______

𝟏

𝟐, because 𝟓

𝟏

𝟐 = √𝟓

xii. 𝐖𝐡𝐚𝐭𝐏𝐨𝐰𝐞𝐫𝟏

𝟐

(𝟏𝟖

) = ______

𝟑, because (𝟏𝟐

)𝟑

=𝟏𝟖

xiii. 𝐖𝐡𝐚𝐭𝐏𝐨𝐰𝐞𝐫𝟒𝟐(𝟏) = ______

𝟎, because 𝟒𝟐𝟎 = 𝟏

xiv. 𝐖𝐡𝐚𝐭𝐏𝐨𝐰𝐞𝐫𝟏𝟎𝟎(𝟎. 𝟎𝟏) = ______

−𝟏, because 𝟏𝟎𝟎−𝟏 = 𝟎. 𝟎𝟏

xv. 𝐖𝐡𝐚𝐭𝐏𝐨𝐰𝐞𝐫𝟐 (𝟏𝟒

) = ______

−𝟐, because 𝟐−𝟐 =𝟏𝟒






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xvi. 𝐖𝐡𝐚𝐭𝐏𝐨𝐰𝐞𝐫𝟏

𝟒

(𝟐) = ______

−𝟏𝟐, because (

𝟏

𝟒)

−𝟏

𝟐= 𝟒

𝟏

𝟐 = 𝟐

b. With your group members, write a definition for the function 𝐖𝐡𝐚𝐭𝐏𝐨𝐰𝐞𝐫𝒃, where 𝒃 is a number.

The value of 𝐖𝐡𝐚𝐭𝐏𝐨𝐰𝐞𝐫𝒃 is the number you need to raise 𝒃 to in order to get 𝒙. That is, if 𝒃𝑳 = 𝒙, then

𝑳 = 𝐖𝐡𝐚𝐭𝐏𝐨𝐰𝐞𝐫𝒃(𝒙).

Discussion (3 minutes)

Discuss the definitions students created, but do not settle on an official definition just yet. To reinforce the idea of how

this function works, ask students a series of “WhatPower” questions, writing the expressions on the board or the

document camera and reading WhatPower𝑏(𝑥) as “What power of 𝑏 is 𝑥?” Be sure that students are visually seeing the

odd structure of this notation and hearing the question “What power of 𝑏 is 𝑥?” to reinforce the meaning of this

function that depends on both the parameter 𝑏 and the variable 𝑥.

WhatPower2(16)

4

WhatPower2(4)

2

WhatPower2(√2)

1

2

WhatPower2(1)

0

WhatPower2 (18

)

−3

Exercises 1–9 (8 minutes)

The point of this set of exercises is for students to determine which real numbers 𝑏 make sense as a base for the

WhatPower𝑏 function. Have students complete this exercise in pairs or small groups, and allow time for students to

debate.

Exercises 1–9

Evaluate the following expressions, and justify your answers.

1. 𝐖𝐡𝐚𝐭𝐏𝐨𝐰𝐞𝐫𝟕(𝟒𝟗)

𝐖𝐡𝐚𝐭𝐏𝐨𝐰𝐞𝐫𝟕(𝟒𝟗) = 𝟐 because 𝟕𝟐 = 𝟒𝟗.

2. 𝐖𝐡𝐚𝐭𝐏𝐨𝐰𝐞𝐫𝟎(𝟕)

𝐖𝐡𝐚𝐭𝐏𝐨𝐰𝐞𝐫𝟎(𝟕) does not make sense because there is no power of 𝟎 that will produce 𝟕.

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3. 𝐖𝐡𝐚𝐭𝐏𝐨𝐰𝐞𝐫𝟓(𝟏)

𝐖𝐡𝐚𝐭𝐏𝐨𝐰𝐞𝐫𝟓(𝟏) = 𝟎 because 𝟓𝟎 = 𝟏.

4. 𝐖𝐡𝐚𝐭𝐏𝐨𝐰𝐞𝐫𝟏(𝟓)

𝐖𝐡𝐚𝐭𝐏𝐨𝐰𝐞𝐫𝟏(𝟓) does not exist because for any exponent 𝑳, 𝟏𝑳 = 𝟏, so there is no power of 𝟏 that will produce 𝟓.

5. 𝐖𝐡𝐚𝐭𝐏𝐨𝐰𝐞𝐫−𝟐(𝟏𝟔)

𝐖𝐡𝐚𝐭𝐏𝐨𝐰𝐞𝐫−𝟐(𝟏𝟔) = 𝟒 because (−𝟐)𝟒 = 𝟏𝟔.

6. 𝐖𝐡𝐚𝐭𝐏𝐨𝐰𝐞𝐫−𝟐(𝟑𝟐)

𝐖𝐡𝐚𝐭𝐏𝐨𝐰𝐞𝐫−𝟐(𝟑𝟐) does not make sense because there is no power of −𝟐 that will produce 𝟑𝟐.

7. 𝐖𝐡𝐚𝐭𝐏𝐨𝐰𝐞𝐫𝟏

𝟑

(𝟗)

𝐖𝐡𝐚𝐭𝐏𝐨𝐰𝐞𝐫𝟏

𝟑

(𝟗) = −𝟐 because (𝟏𝟑

)−𝟐

= 𝟗.

8. 𝐖𝐡𝐚𝐭𝐏𝐨𝐰𝐞𝐫−

𝟏

𝟑

(𝟐𝟕)

𝐖𝐡𝐚𝐭𝐏𝐨𝐰𝐞𝐫−

𝟏

𝟑

(𝟐𝟕) does not make sense because there is no power of −𝟏𝟑

that will produce 𝟐𝟕.

9. Describe the allowable values of 𝒃 in the expression 𝐖𝐡𝐚𝐭𝐏𝐨𝐰𝐞𝐫𝒃(𝒙). When can we define a function

𝒇(𝒙) = 𝐖𝐡𝐚𝐭𝐏𝐨𝐰𝐞𝐫𝒃(𝒙)? Explain how you know.

If 𝒃 = 𝟎 or 𝒃 = 𝟏, then the expression 𝐖𝐡𝐚𝐭𝐏𝐨𝐰𝐞𝐫𝒃(𝒙) does not make sense. If 𝒃 < 𝟎, then the expression

𝐖𝐡𝐚𝐭𝐏𝐨𝐰𝐞𝐫𝒃(𝒙) makes sense for some values of 𝒙 but not for others, so we cannot define a function

𝒇(𝒙) = 𝐖𝐡𝐚𝐭𝐏𝐨𝐰𝐞𝐫𝒃(𝒙) if 𝒃 < 𝟎. Thus, we can define the function 𝒇(𝒙) = 𝐖𝐡𝐚𝐭𝐏𝐨𝐰𝐞𝐫𝒃(𝒙) if 𝒃 > 𝟎 and 𝒃 ≠ 𝟏.

Discussion (5 minutes)

Ask student groups to share their responses to Exercise 9 in which they determined which values of 𝑏 are allowable in

the WhatPower𝑏 function. By the end of this Discussion, be sure that all groups understand that it is necessary to

restrict 𝑏 so that either 0 < 𝑏 < 1 or 𝑏 > 1. Then, continue on to rename the WhatPower function to its true name, the

logarithm base 𝑏.

What we are calling the “WhatPower” function is known by the mathematical term logarithm, built from the

Greek word logos (pronounced lo-gohs), meaning ratio, and arithmos (pronounced uh-rith-mohs), meaning

number. The number 𝑏 is the base of the logarithm, and we denote the logarithm base 𝑏 of 𝑥 (which means

the power to which we raise 𝑏 to get 𝑥) by log𝑏(𝑥). That is, whenever you see log𝑏(𝑥), think of

WhatPower𝑏(𝑥).






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Discuss the definition shown in the Frayer diagram below. Ask students to articulate the definition in their own words to

a partner and then share some responses. Have students work with a partner to fill in the remaining parts of the

diagram and then share responses as a class. Provide some sample examples and non-examples as needed to illustrate

some of the characteristics of logarithms.

What are some examples of logarithms?

log2(4) = 2

log3(27) = 3

log10(0.10) = −1

What are some non-examples?

log0(4)

log1(4)

Why can’t 𝑏 = 0? Why can’t 𝑏 = 1?

If there is a number 𝐿 so that log0(4) = 𝐿, then 0𝐿 = 4. But there is no number 𝐿 such that 0𝐿 is 4, so

this does not make sense. Similar reasoning can be applied to log1(4).

Is log5(25) a valid example?

Yes. log5(25) = 2 because 52 = 25.

Is log5(−25) a valid example?

No. There is no number 𝐿 such that 5𝐿 = −25. It is impossible to raise a positive base to an exponent

and get a negative value.

Is log5(0) a valid example?

No. There is no number 𝐿 such that 5𝐿 = 0. It is impossible to raise a positive base to an exponent and

get an answer of 0.

So, what are some characteristics of logarithms?

The base 𝑏 must be a positive number not equal to 1. The input must also be a positive number. The

output may be any real number (positive, negative, or 0).

Logarithm

Definition of the Logarithm Base 𝒃

If three numbers 𝑳, 𝒃, and 𝒙 with 𝟎 < 𝒃 < 𝟏

or 𝒃 > 𝟏 are related by 𝒃𝑳 = 𝒙, then 𝑳 is

the logarithm base 𝒃 of 𝒙, and we write

𝐥𝐨𝐠𝒃(𝒙) = 𝑳.

Characteristics

If 𝒃 > 𝟏 and 𝒙 > 𝟏, 𝐥𝐨𝐠𝒃(𝒙) > 𝟎.

If 𝟎 < 𝒃 < 𝟏 and 𝒙 > 𝟏, then

𝐥𝐨𝐠𝒃(𝒙) < 𝟎.

𝐥𝐨𝐠𝟐(𝟖) = 𝟑

𝐥𝐨𝐠𝟓(𝟏) = 𝟎

𝐥𝐨𝐠𝟏𝟎(𝟎.𝟎𝟏) = −𝟐

Examples

𝐥𝐨𝐠𝟎(𝒙)

𝐥𝐨𝐠−𝟏(𝒙)

𝐥𝐨𝐠𝟏𝟎(−𝟐)

Non-Examples

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Examples (4 minutes)

Lead the class through the computation of the following logarithms. These have all been computed in the Opening

Exercise using the “WhatPower” terminology.

Examples

1. 𝐥𝐨𝐠𝟐(𝟖) = 𝟑

𝟑, because 𝟐𝟑 = 𝟖

2. 𝐥𝐨𝐠𝟑(𝟗) = 𝟐

𝟐, because 𝟑𝟐 = 𝟗

3. 𝐥𝐨𝐠𝟔(𝟑𝟔) = ______

𝟐, because 𝟔𝟐 = 𝟑𝟔

4. 𝐥𝐨𝐠𝟐(𝟑𝟐) =______

𝟓, because 𝟐𝟓 = 𝟑𝟐

5. 𝐥𝐨𝐠𝟏𝟎(𝟏𝟎𝟎𝟎) = ______

𝟑, because 𝟏𝟎𝟑 = 𝟏𝟎𝟎𝟎

6. 𝐥𝐨𝐠𝟒𝟐(𝟏) =______

𝟎, because 𝟒𝟐𝟎 = 𝟏

7. 𝐥𝐨𝐠𝟏𝟎𝟎(𝟎. 𝟎𝟏) =______

−𝟏, because 𝟏𝟎𝟎−𝟏 = 𝟎. 𝟎𝟏

8. 𝐥𝐨𝐠𝟐 (𝟏𝟒

) = ______

−𝟐, because 𝟐−𝟐 =𝟏𝟒

Exercise 10 (6 minutes)

Have students complete this exercise alone or in pairs.

Exercise 10

10. Compute the value of each logarithm. Verify your answers using an exponential statement.

a. 𝐥𝐨𝐠𝟐(𝟑𝟐)

𝐥𝐨𝐠𝟐(𝟑𝟐) = 𝟓 because 𝟐𝟓 = 𝟑𝟐.

b. 𝐥𝐨𝐠𝟑(𝟖𝟏)

𝐥𝐨𝐠𝟑(𝟖𝟏) = 𝟒 because 𝟑𝟒 = 𝟖𝟏.

c. 𝐥𝐨𝐠𝟗(𝟖𝟏)

𝐥𝐨𝐠𝟗(𝟖𝟏) = 𝟐 because 𝟗𝟐 = 𝟖𝟏.

d. 𝐥𝐨𝐠𝟓(𝟔𝟐𝟓)

𝐥𝐨𝐠𝟓(𝟔𝟐𝟓) = 𝟒 because 𝟓𝟒 = 𝟔𝟐𝟓.

e. 𝐥𝐨𝐠𝟏𝟎(𝟏, 𝟎𝟎𝟎, 𝟎𝟎𝟎, 𝟎𝟎𝟎)

𝐥𝐨𝐠𝟏𝟎(𝟏, 𝟎𝟎𝟎, 𝟎𝟎𝟎, 𝟎𝟎𝟎) = 𝟗,

because 𝟏𝟎𝟗 = 𝟏, 𝟎𝟎𝟎, 𝟎𝟎𝟎, 𝟎𝟎𝟎.

f. 𝐥𝐨𝐠𝟏𝟎𝟎𝟎(𝟏𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎)

𝐥𝐨𝐠𝟏𝟎𝟎𝟎(𝟏𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎) = 𝟑, because

𝟏𝟎𝟎𝟎𝟑 = 𝟏, 𝟎𝟎𝟎, 𝟎𝟎𝟎, 𝟎𝟎𝟎.

g. 𝐥𝐨𝐠𝟏𝟑(𝟏𝟑)

𝐥𝐨𝐠𝟏𝟑(𝟏𝟑) = 𝟏 because 𝟏𝟑𝟏 = 𝟏𝟑.

h. 𝐥𝐨𝐠𝟏𝟑(𝟏)

𝐥𝐨𝐠𝟏𝟑(𝟏) = 𝟎 because 𝟏𝟑𝟎 = 𝟏.

Scaffolding:

If students are struggling

with notation, give them

examples where they

convert between

logarithmic and

exponential form.

Use this chart as a visual

support.

Logarithmic

Form

Exponential

Form

log8(64) = 2

8−2 =1

64

log64(4) =1

3






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i. 𝐥𝐨𝐠𝟕(√𝟕)

𝐥𝐨𝐠𝟕(√𝟕) =𝟏𝟐

because 𝟕𝟏

𝟐 = √𝟕.

j. 𝐥𝐨𝐠𝟗(𝟐𝟕)

𝐥𝐨𝐠𝟗(𝟐𝟕) =𝟑𝟐

because 𝟗𝟑

𝟐 = 𝟑𝟑 = 𝟐𝟕

k. 𝐥𝐨𝐠√𝟕(𝟕)

𝐥𝐨𝐠√𝟕(𝟕) = 𝟐 because (√𝟕)𝟐

= 𝟕.

l. 𝐥𝐨𝐠√𝟕 (𝟏

𝟒𝟗)

𝐥𝐨𝐠√𝟕 (𝟏

𝟒𝟗) = −𝟒 because (√𝟕)

−𝟒=

𝟏

(√𝟕)𝟒 =

𝟏𝟒𝟗

.

m. 𝐥𝐨𝐠𝒙(𝒙𝟐)

𝐥𝐨𝐠𝒙(𝒙𝟐) = 𝟐 because (𝒙)𝟐 = 𝒙𝟐.

Closing (2 minutes)

Ask students to summarize the important parts of the lesson, either in writing, to a partner, or as a class. Use this as an

opportunity to informally assess understanding of the lesson. The following are some important summary elements.

Exit Ticket (5 minutes)

Lesson Summary

If three numbers 𝑳, 𝒃, and 𝒙 are related by 𝒙 = 𝒃𝑳, then 𝑳 is the logarithm base 𝒃 of 𝒙, and we write

𝐥𝐨𝐠𝒃(𝒙) = 𝑳. That is, the value of the expression 𝐥𝐨𝐠𝒃(𝒙) is the power of 𝒃 needed to obtain 𝒙.

Valid values of 𝒃 as a base for a logarithm are 𝟎 < 𝒃 < 𝟏 and 𝒃 > 𝟏.






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Name Date


Exit Ticket

1. Explain why we need to specify 0 < 𝑏 < 1 and 𝑏 > 1 as valid values for the base 𝑏 in the expression log𝑏(𝑥).

2. Calculate the following logarithms.

a. log5(25)

b. log10 (1

100)

c. log9(3)






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Exit Ticket Sample Solutions

1. Explain why we need to specify 𝟎 < 𝒃 < 𝟏 and 𝒃 > 𝟏 as valid values for the base 𝒃 in the expression 𝐥𝐨𝐠𝒃(𝒙).

If 𝒃 = 𝟎, then 𝐥𝐨𝐠𝟎(𝒙) = 𝑳 means that 𝟎𝑳 = 𝒙, which cannot be true if 𝒙 ≠ 𝟎.

If 𝒃 = 𝟏, then 𝐥𝐨𝐠𝟏(𝒙) = 𝑳 means that 𝟏𝑳 = 𝒙, which cannot be true if 𝒙 ≠ 𝟏.

If 𝒃 < 𝟎, then 𝐥𝐨𝐠𝐛(𝒙) = 𝑳 makes sense for some but not all values of 𝒙 > 𝟎; for example, if 𝒃 = −𝟐 and 𝒙 = 𝟑𝟐, there is no power of −𝟐 that would produce 𝟑𝟐, so 𝐥𝐨𝐠−𝟐(𝟑𝟐) does not makes sense.

Thus, if 𝒃 ≤ 𝟎 or 𝒃 = 𝟏, then for many values of 𝒙, the expression 𝐥𝐨𝐠𝒃(𝒙) does not make sense.

2. Calculate the following logarithms.

a. 𝐥𝐨𝐠𝟓(𝟐𝟓)

𝐥𝐨𝐠𝟓(𝟐𝟓) = 𝟐

b. 𝐥𝐨𝐠𝟏𝟎 (𝟏

𝟏𝟎𝟎)

𝐥𝐨𝐠𝟏𝟎 (𝟏

𝟏𝟎𝟎) = −𝟐

c. 𝐥𝐨𝐠𝟗(𝟑)

𝐥𝐨𝐠𝟗(𝟑) =𝟏

𝟐

Problem Set Sample Solutions

In this introduction to logarithms, students are only asked to find simple logarithms base 𝑏 in which the logarithm is an

integer or a simple fraction and the expression can be calculated by inspection.

1. Rewrite each of the following in the form 𝐖𝐡𝐚𝐭𝐏𝐨𝐰𝐞𝐫𝒃(𝒙) = 𝑳.

a. 𝟑𝟓 = 𝟐𝟒𝟑 b. 𝟔−𝟑 =𝟏

𝟐𝟏𝟔 c. 𝟗𝟎 = 𝟏

𝐖𝐡𝐚𝐭𝐏𝐨𝐰𝐞𝐫𝟑(𝟐𝟒𝟑) = 𝟓 𝐖𝐡𝐚𝐭𝐏𝐨𝐰𝐞𝐫𝟔 (𝟏

𝟐𝟏𝟔) = −𝟑 𝐖𝐡𝐚𝐭𝐏𝐨𝐰𝐞𝐫𝟗(𝟏) = 𝟎

2. Rewrite each of the following in the form 𝐥𝐨𝐠𝒃(𝒙) = 𝑳.

a. 𝟏𝟔𝟏

𝟒 = 𝟐 b. 𝟏𝟎𝟑 = 𝟏, 𝟎𝟎𝟎 c. 𝒃𝒌 = 𝒓

𝐥𝐨𝐠𝟏𝟔(𝟐) =𝟏

𝟒 𝐥𝐨𝐠𝟏𝟎(𝟏, 𝟎𝟎𝟎) = 𝟑 𝐥𝐨𝐠𝒃(𝒓) = 𝒌

3. Rewrite each of the following in the form 𝒃𝑳 = 𝒙.

a. 𝐥𝐨𝐠𝟓(𝟔𝟐𝟓) = 𝟒 b. 𝐥𝐨𝐠𝟏𝟎(𝟎. 𝟏) = −𝟏 c. 𝐥𝐨𝐠𝟐𝟕𝟗 =𝟐𝟑

𝟓𝟒 = 𝟔𝟐𝟓 𝟏𝟎−𝟏 = 𝟎. 𝟏 𝟐𝟕𝟐𝟑 = 𝟗






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4. Consider the logarithms base 𝟐. For each logarithmic expression below, either calculate the value of the expression

or explain why the expression does not make sense.

a. 𝐥𝐨𝐠𝟐(𝟏𝟎𝟐𝟒)

𝟏𝟎

b. 𝐥𝐨𝐠𝟐(𝟏𝟐𝟖)

𝟕

c. 𝐥𝐨𝐠𝟐(√𝟖)

𝟑

𝟐

d. 𝐥𝐨𝐠𝟐 (𝟏

𝟏𝟔)

−𝟒

e. 𝐥𝐨𝐠𝟐(𝟎)

This does not make sense. There is no value of 𝑳 so that 𝟐𝑳 = 𝟎.

f. 𝐥𝐨𝐠𝟐 (−𝟏

𝟑𝟐)

This does not make sense. There is no value of 𝑳 so that 𝟐𝑳 is negative.

5. Consider the logarithms base 𝟑. For each logarithmic expression below, either calculate the value of the expression


a. 𝐥𝐨𝐠𝟑(𝟐𝟒𝟑)

𝟓

b. 𝐥𝐨𝐠𝟑(𝟐𝟕)

𝟑

c. 𝐥𝐨𝐠𝟑(𝟏)

𝟎

d. 𝐥𝐨𝐠𝟑 (𝟏𝟑

)

−𝟏

e. 𝐥𝐨𝐠𝟑(𝟎)

This does not make sense. There is no value of 𝑳 so that 𝟑𝑳 = 𝟎.

f. 𝐥𝐨𝐠𝟑 (−𝟏𝟑

)

This does not make sense. There is no value of 𝑳 so that 𝟑𝑳 < 𝟎.






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6. Consider the logarithms base 𝟓. For each logarithmic expression below, either calculate the value of the expression


a. 𝐥𝐨𝐠𝟓(𝟑𝟏𝟐𝟓)

𝟓

b. 𝐥𝐨𝐠𝟓(𝟐𝟓)

𝟐

c. 𝐥𝐨𝐠𝟓(𝟏)

𝟎

d. 𝐥𝐨𝐠𝟓 (𝟏

𝟐𝟓)

−𝟐

e. 𝐥𝐨𝐠𝟓(𝟎)

This does not make sense. There is no value of 𝑳 so that 𝟓𝑳 = 𝟎.

f. 𝐥𝐨𝐠𝟓 (−𝟏

𝟐𝟓)

This does not make sense. There is no value of 𝑳 so that 𝟓𝑳 is negative.

7. Is there any positive number 𝒃 so that the expression 𝐥𝐨𝐠𝒃(𝟎) makes sense? Explain how you know.

No, there is no value of 𝑳 so that 𝒃𝑳 = 𝟎. I know 𝒃 has to be a positive number. A positive number raised to an

exponent never equals 𝟎.

8. Is there any positive number 𝒃 so that the expression 𝐥𝐨𝐠𝒃(−𝟏) makes sense? Explain how you know.

No. Since 𝒃 is positive, there is no value of 𝑳 so that 𝒃𝑳 is negative. A positive number raised to an exponent never

has a negative value.

9. Verify each of the following by evaluating the logarithms.

a. 𝐥𝐨𝐠𝟐(𝟖) + 𝐥𝐨𝐠𝟐(𝟒) = 𝐥𝐨𝐠𝟐(𝟑𝟐)

𝟑 + 𝟐 = 𝟓

b. 𝐥𝐨𝐠𝟑(𝟗) + 𝐥𝐨𝐠𝟑(𝟗) = 𝐥𝐨𝐠𝟑(𝟖𝟏)

𝟐 + 𝟐 = 𝟒

c. 𝐥𝐨𝐠𝟒(𝟒) + 𝐥𝐨𝐠𝟒(𝟏𝟔) = 𝐥𝐨𝐠𝟒(𝟔𝟒)

𝟏 + 𝟐 = 𝟑

d. 𝐥𝐨𝐠𝟏𝟎(𝟏𝟎𝟑) + 𝐥𝐨𝐠𝟏𝟎(𝟏𝟎𝟒) = 𝐥𝐨𝐠𝟏𝟎(𝟏𝟎𝟕)

𝟑 + 𝟒 = 𝟕

10. Looking at the results from Problem 9, do you notice a trend or pattern? Can you make a general statement about

the value of 𝐥𝐨𝐠𝒃(𝒙) + 𝐥𝐨𝐠𝒃(𝒚)?

The sum of two logarithms of the same base is found by multiplying the input values,

𝐥𝐨𝐠𝒃(𝒙) + 𝐥𝐨𝐠𝒃(𝒚) = 𝐥𝐨𝐠𝒃(𝒙𝒚). (Note to teacher: Do not evaluate this answer harshly. This is just a preview of a

property that students learn later in the module.)






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11. To evaluate 𝐥𝐨𝐠𝟐(𝟑), Autumn reasoned that since 𝐥𝐨𝐠𝟐(𝟐) = 𝟏 and 𝐥𝐨𝐠𝟐(𝟒) = 𝟐, 𝐥𝐨𝐠𝟐(𝟑) must be the average of 𝟏

and 𝟐 and therefore 𝐥𝐨𝐠𝟐(𝟑) = 𝟏. 𝟓. Use the definition of logarithm to show that 𝐥𝐨𝐠𝟐(𝟑) cannot be 𝟏. 𝟓. Why is

her thinking not valid?

According to the definition of logarithm, 𝐥𝐨𝐠𝟐(𝟑) = 𝟏. 𝟓 only if 𝟐𝟏.𝟓 = 𝟑. According to the calculator, 𝟐𝟏.𝟓 ≈ 𝟐. 𝟖𝟐𝟖,

so 𝐥𝐨𝐠𝟐(𝟑) cannot be 𝟏. 𝟓. Autumn was assuming that the outputs would follow a linear pattern, but since the

outputs are exponents, the relationship is not linear.

12. Find the value of each of the following.

a. If 𝒙 = 𝐥𝐨𝐠𝟐(𝟖) and 𝒚 = 𝟐𝒙, find the value of 𝒚.

𝒚 = 𝟖

b. If 𝐥𝐨𝐠𝟐(𝒙) = 𝟔, find the value of 𝒙.

𝒙 = 𝟔𝟒

c. If 𝒓 = 𝟐𝟔 and 𝒔 = 𝐥𝐨𝐠𝟐(𝒓), find the value of 𝒔.

𝒔 = 𝟔





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