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Unit 4, Lesson 10

Interpreting and Writing Logarithmic Equations

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Integrated Math 3

10.1

Warm-up

Standards Alignment

Building On

Addressing

Instructional Routines

Take Turns

Materials

None

Activity Narrative

In this Warm-up, students work in groups of 2 and take turns reading and interpreting logarithmic equations. Although the logarithms students have seen so far are in base 10, here they encounter simple logarithms in base 2 and base 5, which prompt them to transfer their understanding about the structure of base-10 logarithms to these other bases.

Launch

Arrange students in groups of 2. Ask them to take turns reading and interpreting to each other the logarithmic expressions in the activity.

Activity

None

The expression can be read as: “The log, base 10, of 1,000 is 3.”

It can be interpreted as: “The exponent to which we raise a base 10 to get 1,000 is 3.”

Take turns with a partner reading each equation out loud. Then, interpret what they mean.

Student Response

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Building on Student Thinking

Activity Synthesis

Invite students to share how they thought about reading each equation, focusing on the equations in base 2 and base 5. Some students may notice that those log equations share the same structure as those in base 10 and make use of that structure to interpret the equations. Highlight their observations.

10.2

Activity

Instructional Routines

MLR5: Co-Craft Questions

Materials

None

Activity Narrative

This activity extends students’ understanding of logarithms to include logarithms in another base. Students analyze patterns in a base-2 logarithm table and notice that it can be interpreted the same way as the base-10 table, except the values in the right-hand column are the exponents in expressions with a base of 2 (MP7).

Students use the table to evaluate base-2 log expressions and to solve simple exponential equations in base 2. They continue to reason abstractly and quantitatively (MP6) about the meaning of each parameter in the equations.

The work here prepares students to see an equation in the form of and one in the form of as equivalent, an understanding that they will develop in the next activity.

This activity uses the Co-Craft Questions math language routine to advance reading and writing as students make sense of a context and practice generating mathematical questions.

10.3

Activity

Instructional Routines

MLR8: Discussion Supports

Materials

None

Activity Narrative

In this activity, students make explicit connections between equivalent equations in exponential form and in logarithmic form. They also write equations to represent descriptions of exponential relationships. Working across different forms and representations reinforces students’ understanding of the meaning of logarithms. Along the way, they continue to attend carefully to the meaning of each parameter in the equations they write (MP6).

After converting a series of numerical equations from one form to the other, students generalize the equivalence of the two forms algebraically (MP8). Note that there are restrictions on the parameter : the base cannot be negative, and we don’t typically look at bases that are less than 1. Students don’t need to concern themselves with this at the moment, however.

The activity includes equations in various bases, but students should not be assessed on their readiness to work in bases other than 2 and 10.

Lesson Synthesis

To reinforce the key ideas in the lesson, discuss questions such these:

“How are , , and alike? How are they different?” (They all represent the exponent of a power expression that has a value of 60, and they are all logarithms. They express the exponent for powers with different bases.)
“Is the equation true? How do you know?” (Yes. The 100 is the base, the is the exponent, and 10 is what we get if we raise 100 to exponent , and .)
“How might we write the solution to ?” ( or 5.1699)
“Does it matter if we write the solution as or 5.1699? Is there a difference?” (The solution written as a logarithm is exact. The other solution is an approximation.)
“We learned that and are equivalent. How would you explain to someone why we consider these two equations as equivalent, even though they don’t look alike?” (They are two ways to represent the same relationship. tells us the exponent by which we raise a base of 7 to get 49, and that exponent is 2.)
“Suppose some positive number raised to exponent has a value of 200. How can we express this relationship in exponential form and logarithmic form?” ( and )

to access Cool-down.

Student Lesson Summary

Many relationships that can be expressed with an exponent can also be expressed with a logarithm. Let’s look at this equation: The base is 2 and the exponent is 7, so it can be expressed as a logarithm with base 2:

In general, an exponential equation and a logarithmic equation are related as shown here:

<p>Two equations, annotated. B to the y power = x, b is the base, y is the exponent. Second equation, log, subscript b, x = y. B is indicated to be the base, and an arrow indicates y as the exponent.</p>

Exponents can be negative, so a logarithm can have negative values. For example, , which means that .

An exponential equation cannot always be solved by observation. For example, does not have an obvious solution. The logarithm gives us a way to represent the solution to this equation: . The expression is approximately 4.2479, but is an exact solution.

Launch

Arrange students in groups of 2. Introduce the context of logarithms with a base of 2. Use Co-Craft Questions to orient students to the context and to elicit possible mathematical questions.

Display only the logarithm tables, without revealing the questions. Give students 1–2 minutes to write a list of mathematical questions that could be asked about the situation, before comparing questions with a partner.
Invite several partners to share one question with the class. Record the questions for all to see. Ask the class to make comparisons among the shared questions and their own. Ask, “What do these questions have in common? How are they different?” Listen for and amplify language related to the learning goal, such as “exponent,” “base,” and “evaluate.”
Reveal the instructions, and give students 1–2 minutes to compare them to their own question and those of their classmates. Invite students to identify similarities and differences by asking:
- “Which of your questions is most similar to or different from the instructions provided? Why?”
- “Is there a main mathematical concept that is present in both your questions and the instructions provided? If so, describe it.”

Ask students to discuss with a partner why it makes sense that has a value of 3 and has a value of 0. Before students start working on the task, ensure that they can articulate that has a value of 3 because and has a value of 0 because .

Give students quiet work time and then time to share their work with a partner. Listen for students who can articulate the meanings of the logarithms, and ask them to share during the whole-class discussion.

Activity

None


1	0
2	1
3	1.5850
4	2
5	2.3219
6	2.5850
7	2.8074
8	3
9	3.1699
10	3.3219


11	3.4594
12	3.5845
13	3.7004
14	3.8074
15	3.9069
16	4
17	4.0875
18	4.1699
19	4.2479
20	4.3219


21	4.3923
22	4.4594
23	4.5236
24	4.5850
25	4.6439
26	4.7004
27	4.7549
28	4.8074
29	4.8580
30	4.9069


31	4.9542
32	5
33	5.0444
34	5.0875
35	5.1293
36	5.1699
37	5.2095
38	5.2479
39	5.2854
40	5.3219

Use the tables to find the exact or approximate value of each expression. Then explain to a partner what each expression and its approximated value means.
Solve each equation. Write the solution as a logarithmic expression.

Student Response

to access Student Response.

Building on Student Thinking

If students are not sure where to start when asked to solve the equations, consider saying:

“Tell me more about the expressions in the Warm-up and what they mean.”
“How does the expression relate to the exponent that 2 is raised to?”

Launch

Display these two equations for all to see: Tell students that the first equation is written in exponential form and the second equation is in logarithmic form.

Explain that the two equations show the same information in two different ways. They represent the same relationship between a base, an exponent, and the value of that base after it is raised to the exponent. Consider articulating the meaning of each equation verbally:

can be interpreted as: “Raising 2 to exponent 3 gives 8.”
can be interpreted as: “The exponent to which we raise a base 2 to get 8 is 3.”

Activity

None

These equations express the same relationship between 2, 16, and 4:

Each row shows two equations that express the same relationship. Complete the table.

	exponential form	logarithmic form

a.
b.
c.
d.
e.
f.
g.
h.
i.
j.

Write two equations—one in exponential form and one in logarithmic form—to represent each question. Use “?” for the unknown value.
1. “To what exponent do we raise the number 4 to get 64?”
2. “What is the log, base 2, of 128?”

Student Response

to access Student Response.

Building on Student Thinking

If students do not yet correctly write the exponential and corresponding logarithmic equations in the table, consider asking:

“Can you explain how you wrote your equation.”
“What is the same and what is different about the equations and ?”

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Lesson 10

Interpreting and Writing Logarithmic Equations

Warm-up

Standards Alignment

Building On

Addressing

HSF-LE.A.4

Instructional Routines

Materials

Activity Narrative

Launch

Activity

Student Response

Building on Student Thinking

Activity Synthesis

Activity

Instructional Routines

Materials

Activity Narrative

Activity

Instructional Routines

Materials

Activity Narrative

Lesson Synthesis

Student Lesson Summary

Launch

Activity

Student Response

Building on Student Thinking

Activity Synthesis

Launch

Activity

Student Response

Building on Student Thinking

Activity Synthesis

Building Toward

Standards Alignment

Building On

Addressing

HSF-BF.B.4.a

HSF-LE.A.4

Building Toward

Standards Alignment

Building On

Addressing

HSA-SSE.B.3

HSF-LE.A.4

Building Toward