Lesson 9

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 \(2^5\).

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.

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.

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).

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 \(x\).

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. 

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.

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).

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.

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.

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:

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