Lesson 8
NYS COMMON CORE MATHEMATICS CURRICULUM
M3
ALGEBRA II
Lesson 8: The “WhatPower” Function
Student Outcomes

Students calculate a simple logarithm using the definition.
Lesson Notes
The term logarithm is foreign and can be intimidating, so we begin the lesson with a more intuitive function, the
“WhatPower” function, which is a simple renaming of the logarithm 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 just the first introduction to logarithms, and the work done here will prepare students to solve exponential
equations of the form 𝑎𝑏 𝑐𝑡 = 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 will 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, we develop the ideas and notation of logarithmic expressions, 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 is about two minutes left, instruct groups that have not finished parts (a)–(p) to skip to part (q)
so that all groups have time to think about and state the definition of this function. You may choose to collect the
groups’ definitions on paper and share some or all of them with the class using the document camera. We will work on
refining this definition through the lessons; in particular, we are interested in the allowable values of the base 𝑏.
Opening Exercise
Evaluate each expression. The first two have been completed for you.
a.
𝐖𝐡𝐚𝐭𝐏𝐨𝐰𝐞𝐫𝟐 (𝟖) = 𝟑
𝟑, because 𝟐𝟑 = 𝟖
b.
𝐖𝐡𝐚𝐭𝐏𝐨𝐰𝐞𝐫𝟑 (𝟗) = 𝟐
𝟐, because 𝟑𝟐 = 𝟗
c.
𝐖𝐡𝐚𝐭𝐏𝐨𝐰𝐞𝐫𝟔 (𝟑𝟔) = ______
𝟐, because 𝟔𝟐 = 𝟑𝟔
Lesson 8:
Date:
The “WhatPower” Function
7/13/17
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NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 8
M3
ALGEBRA II
d.
𝐖𝐡𝐚𝐭𝐏𝐨𝐰𝐞𝐫𝟐 (𝟑𝟐) = ______
𝟓, because 𝟐𝟓 = 𝟑𝟐
e.
𝐖𝐡𝐚𝐭𝐏𝐨𝐰𝐞𝐫𝟏𝟎 (𝟏𝟎𝟎𝟎) = ______
𝟑, because 𝟏𝟎𝟑 = 𝟏𝟎𝟎𝟎
f.
𝐖𝐡𝐚𝐭𝐏𝐨𝐰𝐞𝐫𝟏𝟎 (𝟏, 𝟎𝟎𝟎, 𝟎𝟎𝟎) = ______
𝟔, because 𝟏𝟎𝟔 = 𝟏, 𝟎𝟎𝟎, 𝟎𝟎𝟎
g.
𝐖𝐡𝐚𝐭𝐏𝐨𝐰𝐞𝐫𝟏𝟎𝟎 (𝟏, 𝟎𝟎𝟎, 𝟎𝟎𝟎) = ______
𝟑, because 𝟏𝟎𝟎𝟑 = 𝟏, 𝟎𝟎𝟎, 𝟎𝟎𝟎
h.
𝐖𝐡𝐚𝐭𝐏𝐨𝐰𝐞𝐫𝟒 (𝟔𝟒) = ______
𝟑, because 𝟒𝟑 = 𝟔𝟒
i.
𝐖𝐡𝐚𝐭𝐏𝐨𝐰𝐞𝐫𝟐 (𝟔𝟒) = ______
𝟔, because 𝟐𝟔 = 𝟔𝟒
j.
𝐖𝐡𝐚𝐭𝐏𝐨𝐰𝐞𝐫𝟗 (𝟑) = ______
𝟏
𝟐
k.
𝐖𝐡𝐚𝐭𝐏𝐨𝐰𝐞𝐫𝟓 (√𝟓) = ______
𝟏
𝟐
l.
𝟏
, because 𝟗𝟐 = 𝟑
𝟏
, because 𝟓𝟐 = √𝟓
𝟏
𝟖
𝐖𝐡𝐚𝐭𝐏𝐨𝐰𝐞𝐫𝟏 ( ) = ______
𝟐
𝟏
𝟐
𝟑
𝟑, because ( ) =
m.
𝟏
𝟖
𝐖𝐡𝐚𝐭𝐏𝐨𝐰𝐞𝐫𝟒𝟐 (𝟏) = ______
𝟎, because 𝟒𝟐𝟎 = 𝟏
n.
𝐖𝐡𝐚𝐭𝐏𝐨𝐰𝐞𝐫𝟏𝟎𝟎 (𝟎. 𝟎𝟏) = ______
−𝟏, because 𝟏𝟎𝟎−𝟏 = 𝟎. 𝟎𝟏
o.
𝟏
𝟒
𝐖𝐡𝐚𝐭𝐏𝐨𝐰𝐞𝐫𝟐 ( ) = ______
−𝟐, because 𝟐−𝟐 =
Lesson 8:
Date:
𝟏
𝟒
The “WhatPower” Function
7/13/17
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Lesson 8
NYS COMMON CORE MATHEMATICS CURRICULUM
M3
ALGEBRA II
𝐖𝐡𝐚𝐭𝐏𝐨𝐰𝐞𝐫𝟏 (𝟐) = ______
p.
𝟒
𝟏
𝟏
𝟐
− , because (
𝟏
𝟏 −𝟐
) = 𝟒𝟐 = 𝟐
𝟒
With your group members, write a definition for the function 𝐖𝐡𝐚𝐭𝐏𝐨𝐰𝐞𝐫𝒃 , where 𝒃 is a number.
q.
MP.8
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 the 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)


WhatPower2 (4)


1
2
WhatPower2 (1)


2
WhatPower2 (√2 )


4
0
1
WhatPower2 ( )
8

−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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Date:
The “WhatPower” Function
7/13/17
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Lesson 8
NYS COMMON CORE MATHEMATICS CURRICULUM
M3
ALGEBRA II
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 (𝟗)
𝟑
𝟏
𝟑
𝐖𝐡𝐚𝐭𝐏𝐨𝐰𝐞𝐫𝟏 (𝟗) = −𝟐 because ( )
𝟑
8.
−𝟐
= 𝟗.
𝐖𝐡𝐚𝐭𝐏𝐨𝐰𝐞𝐫−𝟏 (𝟐𝟕)
𝟑
𝟏
𝟑
𝐖𝐡𝐚𝐭𝐏𝐨𝐰𝐞𝐫−𝟏 (𝟐𝟕) does not make sense because there is no power of − that will produce 𝟐𝟕.
𝟑
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 we need 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 log 𝑏 (𝑥). That is, whenever you see log 𝑏 (𝑥), think of
WhatPower𝑏 (𝑥).
Lesson 8:
Date:
The “WhatPower” Function
7/13/17
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Lesson 8
NYS COMMON CORE MATHEMATICS CURRICULUM
M3
ALGEBRA II
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
MP.8
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.
Definition of the logarithm base 𝒃
Characteristics
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
𝐥𝐨𝐠 𝒃 (𝒙) = 𝑳.
If 𝒃 > 𝟏 and 𝒙 > 𝟏, 𝐥𝐨𝐠 𝒃 (𝒙) > 𝟎.
If 𝟎 < 𝒃 < 𝟏 and 𝒙 > 𝟏, then
𝐥𝐨𝐠 𝒃 (𝒙) < 𝟎.
Logarithm
Examples



𝐥𝐨𝐠 𝟏𝟎(𝟎. 𝟎𝟏) = −𝟐
𝐥𝐨𝐠 𝟏𝟎 (−𝟐)

log 3 (27) = 3

log10 (0.10) = −1
What are some non-examples?

log 0 (4)

log1 (4)
Why can’t 𝑏 = 0? Why can’t 𝑏 = 1?
If there is a number 𝐿 so that log 0 (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 log 5 (25) a valid example?
Yes. log 5 (25) = 2 because 52 = 25.
Is log 5 (−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 log 5 (0) a valid example?


𝐥𝐨𝐠 −𝟏 (𝒙)
log 2 (4) = 2


𝐥𝐨𝐠 𝟎(𝒙)
𝐥𝐨𝐠 𝟓(𝟏) = 𝟎



𝐥𝐨𝐠 𝟐(𝟖) = 𝟑
What are some examples of logarithms?


Non-Examples
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).
Lesson 8:
Date:
The “WhatPower” Function
7/13/17
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Lesson 8
NYS COMMON CORE MATHEMATICS CURRICULUM
M3
ALGEBRA II
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.
𝐥𝐨𝐠 𝟐 (𝟖) = 𝟑
2.
𝟑
𝟐, because 𝟑𝟐 = 𝟗.
𝟑, because 𝟐 = 𝟖.
3.
𝐥𝐨𝐠 𝟔 (𝟑𝟔) = ______
4.
𝟐
𝐥𝐨𝐠 𝟏𝟎 (𝟏𝟎𝟎𝟎) = ______
6.
𝟑, because 𝟏𝟎𝟑 = 𝟏𝟎𝟎𝟎.
7.
𝐥𝐨𝐠 𝟐 (𝟑𝟐) =______
𝟓, because 𝟐𝟓 = 𝟑𝟐.
𝟐, because 𝟔 = 𝟑𝟔.
5.
𝐥𝐨𝐠 𝟑 (𝟗) = 𝟐
𝐥𝐨𝐠 𝟒𝟐 (𝟏) =______
𝟎, because 𝟒𝟐𝟎 = 𝟏.
𝐥𝐨𝐠 𝟏𝟎𝟎 (𝟎. 𝟎𝟏) =______
8.
−𝟏, because 𝟏𝟎𝟎−𝟏 = 𝟎. 𝟎𝟏.
𝟏
𝟒
𝐥𝐨𝐠 𝟐 ( ) = ______
𝟏
𝟒
−𝟐, because 𝟐−𝟐 = .
Exercise 10 (6 minutes)
Have students complete this exercise alone or in pairs.
Scaffolding:
Exercise 10
10. Compute the value of each logarithm. Verify your answers using an exponential statement.
a.
𝐥𝐨𝐠 𝟐 (𝟑𝟐)
b.
𝐥𝐨𝐠 𝟑 (𝟖𝟏) = 𝟒, because 𝟑𝟒 = 𝟖𝟏.
𝐥𝐨𝐠 𝟐 (𝟑𝟐) = 𝟓, because 𝟐𝟓 = 𝟑𝟐.
c.
𝐥𝐨𝐠 𝟗 (𝟖𝟏)
d.
𝐥𝐨𝐠 𝟏𝟎 (𝟏, 𝟎𝟎𝟎, 𝟎𝟎𝟎, 𝟎𝟎𝟎)
f.
𝐥𝐨𝐠 𝟏𝟑 (𝟏𝟑)
h.
𝐥𝐨𝐠 𝟏𝟑 (𝟏𝟑) = 𝟏, because 𝟏𝟑𝟏 = 𝟏𝟑.
𝐥𝐨𝐠 𝟏𝟎𝟎𝟎 (𝟏, 𝟎𝟎𝟎, 𝟎𝟎𝟎, 𝟎𝟎𝟎)
𝐥𝐨𝐠 𝟏𝟎𝟎𝟎 (𝟏, 𝟎𝟎𝟎, 𝟎𝟎𝟎, 𝟎𝟎𝟎) = 𝟑,
because 𝟏𝟎𝟎𝟎𝟑 = 𝟏, 𝟎𝟎𝟎, 𝟎𝟎𝟎, 𝟎𝟎𝟎.
𝐥𝐨𝐠 𝟏𝟎 (𝟏, 𝟎𝟎𝟎, 𝟎𝟎𝟎, 𝟎𝟎𝟎) = 𝟗,
because 𝟏𝟎𝟗 = 𝟏, 𝟎𝟎𝟎, 𝟎𝟎𝟎, 𝟎𝟎.
g.
 Use this chart as a visual
support.
𝐥𝐨𝐠 𝟓 (𝟔𝟐𝟓)
𝐥𝐨𝐠 𝟓 (𝟔𝟐𝟓) = 𝟒, because 𝟓𝟒 = 𝟔𝟐𝟓.
𝐥𝐨𝐠 𝟗 (𝟖𝟏) = 𝟐, because 𝟗𝟐 = 𝟖𝟏.
e.
𝐥𝐨𝐠 𝟑 (𝟖𝟏)
 If students are struggling
with notation, give them
examples where they
convert between
logarithmic and
exponential form.
Logarithmic
form
log 8 (64) = 2
8−2 =
𝐥𝐨𝐠 𝟏𝟑 (𝟏)
𝐥𝐨𝐠 𝟏𝟑 (𝟏) = 𝟎, because 𝟏𝟑𝟎 = 𝟏.
log 64 (4) =
Lesson 8:
Date:
Exponential
form
1
64
1
3
The “WhatPower” Function
7/13/17
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Lesson 8
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M3
ALGEBRA II
𝐥𝐨𝐠 𝟗 (𝟐𝟕)
i.
j.
𝟑
𝟑
𝐥𝐨𝐠 𝟗 (𝟐𝟕) = , because 𝟗𝟐 = 𝟑𝟑 = 𝟐𝟕.
𝟐
𝐥𝐨𝐠 √𝟕 (𝟕)
k.
𝟏
𝟐
𝟏
𝐥𝐨𝐠 𝟕 (√𝟕) = , because 𝟕𝟐 = √𝟕.
l.
𝟐
𝐥𝐨𝐠 √𝟕 (𝟕) = 𝟐, because (√𝟕) = 𝟕.
m.
𝐥𝐨𝐠 𝟕 (√𝟕)
𝟏
)
𝟒𝟗
−𝟒
𝟏
𝟏
𝟏
𝐥𝐨𝐠 √𝟕 ( ) = −𝟒, because (√𝟕) =
𝟒 = 𝟒𝟗.
𝟒𝟗
(√𝟕)
𝐥𝐨𝐠 √𝟕 (
𝐥𝐨𝐠 𝒙 (𝒙𝟐 )
𝐥𝐨𝐠 𝒙 (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.
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
𝐥𝐨𝐠 𝒃 (𝒙). That is, the value of the expression 𝑳 = 𝐥𝐨𝐠 𝒃 (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.
Exit Ticket (5 minutes)
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Date:
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Lesson 8
NYS COMMON CORE MATHEMATICS CURRICULUM
M3
ALGEBRA II
Name
Date
Lesson 8: The “WhatPower” Function
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 log 𝑏 (𝑥).
2.
Calculate the following logarithms.
a.
log 5 (25)
b.
log10 (
c.
log 9 (3)
1
100
)
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Lesson 8
NYS COMMON CORE MATHEMATICS CURRICULUM
M3
ALGEBRA II
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 𝐥𝐨𝐠 𝒃 (𝒙).
If 𝒃 = 𝟎, then 𝐥𝐨𝐠 𝟎 (𝒙) = 𝑳 means that 𝟎𝑳 = 𝒙, which cannot be true if 𝒙 ≠ 𝟎.
If 𝒃 = 𝟏, then 𝐥𝐨𝐠 𝟏 (𝒙) = 𝑳 means that 𝟏𝑳 = 𝒙, which cannot be true if 𝒙 ≠ 𝟏.
If 𝒃 < 𝟎, then 𝐥𝐨𝐠 𝐛 (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 𝐥𝐨𝐠 𝒃 (𝒙) does not make sense.
2.
Calculate the following logarithms.
a.
𝐥𝐨𝐠 𝟓(𝟐𝟓)
𝐥𝐨𝐠 𝟓(𝟐𝟓) = 𝟐
b.
𝐥𝐨𝐠 𝟏𝟎 (
𝟏
)
𝟏𝟎𝟎
𝟏
𝐥𝐨𝐠 𝟏𝟎 (
) = −𝟐
𝟏𝟎𝟎
c.
𝐥𝐨𝐠 𝟗(𝟑)
𝐥𝐨𝐠 𝟗(𝟑) =
𝟏
𝟐
Problem Set Sample Solutions
In this introduction to logarithms, the students are only asked to find simple logarithms base 𝑏, in which the logarithm is
an integer or 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.
c.
𝟗𝟎 = 𝟏
𝐖𝐡𝐚𝐭𝐏𝐨𝐰𝐞𝐫𝟗 (𝟏) = 𝟎
Rewrite each of the following in the form 𝐥𝐨𝐠 𝒃 (𝒙) = 𝑳.
a.
𝟏
b.
𝟏𝟔𝟒 = 𝟐
𝐥𝐨𝐠 𝟏𝟔 (𝟐) =
3.
𝟏
𝟐𝟏𝟔
𝟏
𝐖𝐡𝐚𝐭𝐏𝐨𝐰𝐞𝐫𝟔 (
) = −𝟑
𝟐𝟏𝟔
𝐖𝐡𝐚𝐭𝐏𝐨𝐰𝐞𝐫𝟑 (𝟐𝟒𝟑) = 𝟓
2.
𝟔−𝟑 =
𝟏
𝟒
𝟏𝟎𝟑 = 𝟏, 𝟎𝟎𝟎
c.
𝐥𝐨𝐠 𝟏𝟎 (𝟏, 𝟎𝟎𝟎) = 𝟑
𝒃𝒌 = 𝒓
𝐥𝐨𝐠 𝒃 (𝒓) = 𝒌
Rewrite each of the following in the form 𝒃𝑳 = 𝒙.
a.
𝐥𝐨𝐠 𝟓 (𝟔𝟐𝟓) = 𝟒
b.
𝟓𝟒 = 𝟔𝟐𝟓
Lesson 8:
Date:
𝐥𝐨𝐠 𝟏𝟎 (𝟎. 𝟏) = −𝟏
𝟏𝟎−𝟏 = 𝟎. 𝟏
c.
𝐥𝐨𝐠 𝟐𝟕 𝟗 =
𝟐
𝟑
𝟐
𝟐𝟕𝟑 = 𝟗
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Lesson 8
NYS COMMON CORE MATHEMATICS CURRICULUM
M3
ALGEBRA II
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.
𝐥𝐨𝐠 𝟐 (𝟏𝟎𝟐𝟒)
𝟏𝟎
b.
𝐥𝐨𝐠 𝟐 (𝟏𝟐𝟖)
𝟕
c.
𝐥𝐨𝐠 𝟐 (√𝟖)
𝟑
𝟐
d.
𝐥𝐨𝐠 𝟐 (
𝟏
)
𝟏𝟔
−𝟒
e.
𝐥𝐨𝐠 𝟐 (𝟎)
This does not make sense. There is no value of 𝑳 so that 𝟐𝑳 = 𝟎.
f.
𝐥𝐨𝐠 𝟐 (−
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,
or explain why the expression does not make sense.
a.
𝐥𝐨𝐠 𝟑 (𝟐𝟒𝟑)
𝟓
b.
𝐥𝐨𝐠 𝟑 (𝟐𝟕)
𝟑
c.
𝐥𝐨𝐠 𝟑 (𝟏)
𝟎
d.
𝟏
𝟑
𝐥𝐨𝐠 𝟑 ( )
−𝟏
e.
𝐥𝐨𝐠 𝟑 (𝟎)
This does not make sense. There is no value of 𝑳 so that 𝟑𝑳 = 𝟎.
f.
𝟏
𝟑
𝐥𝐨𝐠 𝟑 (− )
This does not make sense. There is no value of 𝑳 so that 𝟑𝑳 < 𝟎.
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Lesson 8
NYS COMMON CORE MATHEMATICS CURRICULUM
M3
ALGEBRA II
6.
Consider the logarithms base 𝟓. For each logarithmic expression below, either calculate the value of the expression,
or explain why the expression does not make sense.
a.
𝐥𝐨𝐠 𝟓 (𝟑𝟏𝟐𝟓)
𝟓
b.
𝐥𝐨𝐠 𝟓 (𝟐𝟓)
𝟐
c.
𝐥𝐨𝐠 𝟓 (𝟏)
𝟎
d.
𝐥𝐨𝐠 𝟓 (
𝟏
)
𝟐𝟓
−𝟐
e.
𝐥𝐨𝐠 𝟓 (𝟎)
This does not make sense. There is no value of 𝑳 so that 𝟓𝑳 = 𝟎.
f.
𝐥𝐨𝐠 𝟓 (−
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 𝐥𝐨𝐠 𝒃 (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 𝐥𝐨𝐠 𝒃 (−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.
𝐥𝐨𝐠 𝟐 (𝟖) + 𝐥𝐨𝐠 𝟐 (𝟒) = 𝐥𝐨𝐠 𝟐 (𝟑𝟐)
b.
𝟑+𝟐 =𝟓
c.
𝐥𝐨𝐠 𝟑 (𝟗) + 𝐥𝐨𝐠 𝟑 (𝟗) = 𝐥𝐨𝐠 𝟑 (𝟖𝟏)
𝟐+𝟐 =𝟒
𝐥𝐨𝐠 𝟒 (𝟒) + 𝐥𝐨𝐠 𝟒 (𝟏𝟔) = 𝐥𝐨𝐠 𝟒 (𝟔𝟒)
𝟏+𝟐 =𝟑
d.
𝐥𝐨𝐠 𝟏𝟎 (𝟏𝟎𝟑 ) + 𝐥𝐨𝐠 𝟏𝟎 (𝟏𝟎𝟒 ) = 𝐥𝐨𝐠 𝟏𝟎 (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 𝐥𝐨𝐠 𝒃 (𝒙) + 𝐥𝐨𝐠 𝒃 (𝒚)?
The sum of two logarithms of the same base is found by multiplying the input values,
𝐥𝐨𝐠 𝒃 (𝒙) + 𝐥𝐨𝐠 𝒃 (𝒚) = 𝐥𝐨𝐠 𝒃 (𝒙𝒚) (Note to teacher: Do not evaluate this answer harshly. This is just a preview of a
property that students will learn later in the module.)
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NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 8
M3
ALGEBRA II
11. To evaluate 𝐥𝐨𝐠 𝟐 (𝟑), Autumn reasoned that since 𝐥𝐨𝐠 𝟐 (𝟐) = 𝟏 and 𝐥𝐨𝐠 𝟐 (𝟒) = 𝟐, 𝐥𝐨𝐠 𝟐 (𝟑) must be the average of 𝟏
and 𝟐 and, therefore, 𝐥𝐨𝐠 𝟐 (𝟑) = 𝟏. 𝟓. Use the definition of logarithm to show that 𝐥𝐨𝐠 𝟐 (𝟑) cannot be 𝟏. 𝟓. Why is
her thinking not valid?
According to the definition of logarithm, 𝐥𝐨𝐠 𝟐 (𝟑) = 𝟏. 𝟓 only if 𝟐𝟏.𝟓 = 𝟑. According to the calculator, 𝟐𝟏.𝟓 ≈ 𝟐. 𝟖𝟐𝟖,
so 𝐥𝐨𝐠 𝟐 (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 = 𝐥𝐨𝐠 𝟐 (𝟖) and 𝒚 = 𝟐𝒙 , find the value of 𝒚.
𝒚=𝟖
b.
If 𝐥𝐨𝐠 𝟐 (𝒙) = 𝟔, find the value of 𝒙.
𝒙 = 𝟔𝟒
c.
If 𝒓 = 𝟐𝟔 and 𝒔 = 𝐥𝐨𝐠 𝟐 (𝒓), find the value of 𝒔.
𝒔=𝟔
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