Lesson 9

What is a Logarithm?

9.1: Math Talk: Finding Solutions (5 minutes)

Warm-up

The purpose of this Math Talk is to elicit strategies for reasoning about exponential equations, starting with simple equations representing familiar powers of 4, then moving to equations that involve fractional exponents. These strategies will be helpful later in this lesson when students are introduced to logarithms.

In this activity, students have an opportunity to notice and make use of structure (MP7) as they use properties of exponents to reason about expressions. For example, students could find \(4^\frac52\) by viewing it as \(4^\frac42 \boldcdot 4^\frac12\), which is \(16 \boldcdot 2\), or as \(\left(4^\frac12\right)^5\), which is \(2^5\).

Launch

Display one problem at a time. Give students quiet think time for each problem and ask them to give a signal when they have an answer and a strategy. Keep all problems displayed throughout the talk. Follow with a whole-class discussion.

Representation: Internalize Comprehension. To support working memory, provide students with sticky notes or mini whiteboards.
Supports accessibility for: Memory; Organization

Student Facing

Find or estimate the value of each variable mentally.

\(4^a=16\)

\(4^b=2\)

\(4^\frac{5}{2}=c\)

\(4^d=56\)

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

Ask students to share their strategies for each problem. Record and display their responses for all to see. To involve more students in the conversation, consider asking:

  • “Who can restate \(\underline{\hspace{.5in}}\)’s reasoning in a different way?”
  • “Did anyone have the same strategy but would explain it differently?”
  • “Did anyone solve the problem in a different way?”
  • “Does anyone want to add on to \(\underline{\hspace{.5in}}\)’s strategy?”
  • “Do you agree or disagree? Why?”

Tell students that in today’s lesson they will look for another way to closely approximate the solution to an equation like \(4^d=56\). A large range of answers for \(d\) should be expected as students can identify that \(d\) is greater than 2 and less than 3 but further experimentation or calculation is needed to give a more precise estimate.

Speaking: MLR8 Discussion Supports. Display sentence frames to support students when they explain their strategy. For example, “First, I _____ because . . .” or “I noticed _____ so I . . . .” Some students may benefit from the opportunity to rehearse what they will say with a partner before they share with the whole class. 
Design Principle(s): Optimize output (for explanation)

9.2: A Table of Numbers (15 minutes)

Activity

This activity prompts students to observe the numbers in a base-10 logarithm table, introducing them to the idea of logarithms as they make sense of patterns in the table.

Students notice some familiar relationships in pairs of values in the table, which suggest that \(x\) and \(\log_{10}x\) might be related by exponentiation, but in the opposite direction than what they are accustomed to seeing. For example, in the row that shows 100 for \(x\) and 2 for \(\log_{10} x\), the output, 2, is the exponent for \(10^y\) that produces 100. The term “logarithm” is not yet used here and will be introduced in the next activity.

To make sense of the table (especially the many decimals) and to use the table to solve equations, students look for regularity in the numbers and recognize the log values as exponents that produce certain values (MP8).

Launch

Ask students to find the value of \(y\) that makes each equation true:

  • \(10^y = 1\)
  • \(10^y = 10\)
  • \(10^y = 10,\!000\)
  • \(10^y = 100,\!000,\!000\)

Then, ask them to find or estimate the value of \(y\) that makes these equations true:

  • \(10^y = 9\)
  • \(10^y=90\)

Ask a few students to share their estimates and discuss what makes the second set of equations harder to solve. Next, explain that in the past, mathematicians used tables such as the one in the activity to find unknown exponents. Give students a moment to observe the table and confer with a partner about what the table tells us.

Briefly discuss students’ observations before they continue with the rest of the activity. Highlight observations that suggest that the number in the column labeled \(\log_{10} x\) tells us the exponent to use in a power of 10 to get the value of \(x\).

Engagement: Develop Effort and Persistence. Encourage and support opportunities for peer collaboration. When students analyze the table with a partner, display sentence frames to support conversation that develops fluency with the table, such as: “_____ corresponds to _____ ”, “_____ represents _____”, and “If _____ then _____ because . . . .” Invite students to color code the columns to support their thinking visually, and also to annotate the table with their ideas. Encourage students to transfer the information from the table into related forms of exponential equations that are familiar to them, to support their fluency and understanding.
Supports accessibility for: Language; Social-emotional skills
Conversing: MLR2 Collect and Display. Listen for and collect the language students use to make sense of the numbers in the base-10 logarithm table. Write students’ words and phrases on a visual display for all to see, and update it throughout the remainder of the lesson. Remind students to borrow language from the display as needed. This will help students read and use mathematical language.
Design Principle(s): Optimize output (for explanation); Maximize meta-awareness

Student Facing

\(x\) \(\log_{10} (x)\)
2 0.3010
3 0.4771
4 0.6021
5 0.6990
6 0.7782
7 0.8451
8 0.9031
9 0.9542
10 1
\(x\) \(\log_{10} (x)\)
20 1.3010
30 1.4771
40 1.6021
50 1.6990
60 1.7782
70 1.8451
80 1.9031
90 1.9542
100 2
\(x\) \(\log_{10} (x)\)
200 2.3010
300 2.4771
400 2.6021
500 2.6990
600 2.7782
700 2.8451
800 2.9031
900 2.9542
1,000 3
\(x\) \(\log_{10} (x)\)
2,000 3.3010
3,000 3.4771
4,000 3.6021
5,000 3.6990
6,000 3.7782
7,000 3.8451
8,000 3.9031
9,000 3.9542
10,000 4
  1. Analyze the table and discuss with a partner what you think the table tells us.
  2. Use the table to find the value of the unknown exponent that makes each equation true.
    1. \(10^w = 1,\!000\)
    2. \(10^y = 9\)
    3. \(10^z = 90\)
  3. Notice that some values in the columns labeled \(\log_{10} x\) are whole numbers, but most are decimals. Why do you think that is?

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Anticipated Misconceptions

Students may be overwhelmed looking at all the numbers in the table. Encourage these students to look for patterns. For example, the numbers in both columns increase and the decimal part of the logarithms seem to repeat in the different tables. 

Activity Synthesis

Invite students to share how they went about using the table to solve for \(y\). To prompt them to make sense of the solution in each equation, discuss questions such as:

  • “How does the table help you solve \(10^y = 1,\!000\)?” (It tells us that 3 is the number to use for the exponent in \(10^y\) for the expression to equal 1,000.)
  • “The solution to \(10^y = 9\) is 0.9542. What does the solution mean?” (0.9542 is the number to use for the exponent in \(10^y\) for it to equal 9.) “Does that solution make sense? How do we know?” (Yes. We know that \(10^0 =1\) and \(10^1 = 10\), so the exponent must be somewhere in between 0 and 1, likely close to 1, for the power of 10 to equal 9.)

Conclude the discussion by asking students how they could check to see if a solution found using the table is correct (We can enter it in a calculator as the exponent for base 10 and see if it produces the value on the other side of the equation). Show students how to do so by using a calculator to demonstrate that \(10^{1.9542}\) is approximately 90. If students wonder why it is not exactly 90, explain to students that the non-integer values in the table are all approximations rather than exact solutions. They will learn more about how to express the solutions exactly in future lessons.

9.3: Hello, Logarithm! (15 minutes)

Activity

In this activity, students encounter the term logarithm for the first time and try to define it based on the examples they have seen so far. They also see logarithmic equations for the first time and try to interpret each part of the equation. Included in making sense of each part of the logarithmic equations, students consider how to solve for an unknown value in such an equation.

Notice that question marks, instead of variables, are used to represent unknown values in the equations here. This is to reduce the cognitive load for students. Although students have worked with equations with variables for some time now, equations with logarithms look different and more abstract than others they have seen before. It is going to take time and practice for students to intuit what the statements mean. Using variables in different parts of a logarithmic equation at this point may add an unhelpful layer of abstraction.

Interpreting the meaning of each part of a logarithmic equation and articulating how it relates to an exponential equation require precision in thinking and in language (MP6).

Launch

Arrange students in groups of 2. Encourage students to refer to the base-10 logarithm table in the previous activity as needed.

Give students quiet work time and then time to share their work with a partner.

Student Facing

  1. Here are two true equations based on the information from the table:

    \(\begin {align} \log_{10} 100 &= 2\\ \log_{10} 1,\!000 &= 3 \end{align}\)

    What values could replace the “?” in these equations to make them true?

    1. \(\log_{10} 1,\!000,\!000 = {?}\)
    2. \(\log_{10} 1 = {?}\)
    3. \(\log_{10} ? = 4\)
  2. Between which two whole numbers is the value of \(\log_{10} 600\)? Be prepared to explain how do you know.
  3. The term log is short for logarithm. Discuss the following questions with a partner and record your responses:
    1. What do you think logarithm means or does?
    2. Next to “log” is a subscript—a number or letter printed smaller and below the line of text. What do you think the subscript tells us?
    3. What about the other two numbers on either side of the equal sign (for example, the 100 and the 2 in \(\log_{10} 100 = 2\))? What do they tell us?

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Student Facing

Are you ready for more?

  1. For which whole number values of \(n\) is \(\log_{10}(n)\) an integer?
  2. Why will \(\log_{10}(n)\) never be equal to a non-integer rational number?

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Anticipated Misconceptions

Encourage students who struggle to solve the equations to refer to the tables from the previous activity. Bring their attention to the entries that indicate that \(\log_{10}(10) = 1\) and \(\log_{10}(100) = 2\).  The table does not give a value of \(\log_{10}(1)\) but students can make a guess or prediction based on the other values in the table. Encourage students to look for patterns and make their best guess or estimate.

Activity Synthesis

Invite students to share their conjectures on what the term logarithm means and what each part of a logarithmic equation tells us. Refine their definition and interpretation as needed. Make sure students understand that a logarithmic expression is the solution to an exponential equation; it is the exponent that makes the equation valid. In the examples here, the base is 10.

Explain to students that the expression \(\log_{10} 1\) is pronounced “the logarithm, base 10, of 1” or the “the log, base 10, of 1.” Help students generalize the relationship between an exponential expression and a log expression by writing:

\(\begin {align} 10^y &=x\\ \log_{10} x&=y \end{align}\)

Representation: Internalize Comprehension. Use color and annotations to illustrate student thinking. As students share their reasoning about the meaning of the word logarithm and the related representation, scribe their thinking on a visible display. Draw arrows from the representation or a given example to capture student phrases that concisely explain the definition and purpose of each part of the representation.
Supports accessibility for: Visual-spatial processing; Conceptual processing
Speaking: MLR8 Discussion Supports. Use this to routine to provide students with multiple opportunities to produce the language of logarithms. Display the expression, \(\log_{10} 1\) for all to see. Read “the logarithm, base 10, of 1” or the “the log, base 10, of 1” aloud as you point to the appropriate parts of the extression. Invite all students to repeat in unison 1–2 times. Choral repetition facilitates the transfer of new vocabulary to students’ long term memory. 
Design Principle(s): Maximize meta-awareness

Lesson Synthesis

Lesson Synthesis

To help students build their initial understanding of logarithms, ask questions such as:

  • “We saw equations such as \(\log_{10} 100,\!000 = 5\) today. In that equation, what do the 10, 100,000, and 5 tell us?” (10 is the base of an exponential expression, 5 is the exponent, and 100,000 is the result of raising a base 10 to exponent 5.)
  • “How do we read an equation such as \(\log_{10} 100,\!000 = 5\)?” (The log, base 10, of 100,000 is 5.)
  • “How would you describe the meaning of logarithm to someone who has not heard of it?” (Logarithm represents the exponent in an expression that has a certain base and a certain value.)

9.4: Cool-down - Explaining Logarithm to a Friend (5 minutes)

Cool-Down

Cool-downs for this lesson are available at one of our IM Certified Partners

Student Lesson Summary

Student Facing

We know how to solve equations such as \(10^a = 10,\!000\) or \(10^b= \frac{1}{100}\) by thinking about integer powers of 10. The solutions are \(a = 4\) and \(b = {\text-}2\). What about an equation such as \(10^p = 250\)?

Because \(10^2 = 100\) and \(10^3 = 1,\!000\), we know that \(p\) is between 2 and 3. We can use a logarithm to represent the exact solution to this equation and write it as:

\(\displaystyle p=\log_{10} 250\)

The expression is read “the log, base 10, of 250.”

  • The small, slightly lowered “10” refers to the base of 10.
  • The 250 is the value of the power of 10.
  • \(\log_{10} 250\) is the value of the exponent \(p\) that makes \(10^p\) equal 250.

Base 10 logarithms are often written without the number 10. So \(\log_{10} 250\) can also be written as \(\log 250\) and this expression is read “the log of 250.”

One way to estimate logarithms is with a logarithm table. For example, using this base 10 logarithm table we can see that \(\log_{10} 250\) is between 2.3 and 2.48.

\(x\) \(\log_{10} (x)\)
2 0.3010
3 0.4771
4 0.6021
5 0.6990
6 0.7782
7 0.8451
8 0.9031
9 0.9542
10 1
\(x\) \(\log_{10} (x)\)
20 1.3010
30 1.4771
40 1.6021
50 1.6990
60 1.7782
70 1.8451
80 1.9031
90 1.9542
100 2
\(x\) \(\log_{10} (x)\)
200 2.3010
300 2.4771
400 2.6021
500 2.6990
600 2.7782
700 2.8451
800 2.9031
900 2.9542
1,000 3
\(x\) \(\log_{10} (x)\)
2,000 3.3010
3,000 3.4771
4,000 3.6021
5,000 3.6990
6,000 3.7782
7,000 3.8451
8,000 3.9031
9,000 3.9542
10,000 4