Lesson 10

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.

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

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.

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 that this time the values in the right column are the exponents in expressions with base 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 precisely about the meaning of each parameter in the equations (MP6).

The work here prepares students to see an equation in the form of \(b^y=x\) and one in the form of \(\log_b x = y\) as equivalent, an understanding that they will develop in the next activity.

As students discuss their thinking with a partner, monitor for those who can clearly articulate the meanings of given log expressions and equations to share during the whole-class discussion.

Arrange students in groups of 2. Display the base 2 logarithm table for all to see. Ask students to discuss with a partner why it makes sense that \(\log_2 8\)  has a value of 3 and \(\log_2 1\) has a value of 0. Before students start working on the task, ensure they can articulate that \(\log_2 8\) has a value of 3 because \(2^3 = 8\) and \(\log_2 1\) has a value of 0 because \(2^0 = 1\).

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

As in the previous, students may be overwhelmed with the tables. Point out that the second column has the values \(\log_{2}(x)\) and so students need to find the appropriate value of \(x\) in the first column. For students not sure where to start when asked to solve the equations, ask them to return to the work of the warm-up and consider the meaning of \(\log_2(x)\) (the power \(2\) needs to be raised to in order to give a value of \(x\)).

Select previously identified students to share their responses and reasoning. Focus the discussion on two key ideas:

• In the equation \(2^y = 8\), we know that the value of \(y\) is 3 because \(2^3=8\), but we also know that the value of \(y\) is \(\log_2 8\), because by definition, \(\log_2 8\) is the exponent to which 2 should be raised to get 8. So we can express the solution as either 3, or \(\log_2 8\), as the two are equal.
• When solving exponential equations, the solutions expressed in logarithms are exact solutions, whereas those written in decimals are approximations. For example, the solution to \(2^y = 8\) is \(\log_2 8\) or 3. In this case, both are exact solutions. For \(2^y = 5\), the solution \(\log_2 5\) is exact, but the value we see in the base 2 log table, 2.32193, is not. If we raise 2 to the exponent of 2.32193, the result is not exactly 5.

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 \(b\): 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.

Display the following two equations for all to see: \(\displaystyle 2^3 = 8\) \(\displaystyle \log_2 8 =3\)Tell students that the former is written in exponential form, the latter is in logarithmic form, and the two equations are equivalent.

Explain that the two equations 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:

• \(2^3 = 8\) can be interpreted as: “Raising 2 to exponent 3 gives 8.”
• \(\log_2 8 =3\) can be interpreted as: “The exponent to which we raise a base 2 to get 8 is 3.”

For students who struggle writing the exponential and logarithmic equations, focus their attention on the two given equations \(2^4 = 16\) and \(\log_2(16) = 4\). How are they related? Highlight that the logarithm gives the solution to the exponential equation \(2^? = 16\). Conversely, given an exponential equation like \(2^4 = 16\) this tells us that \(\log_2(16) = 4\) since it gives the exponent 2 needs to be raised to in order to give 16. 

Make sure students understand the connections between the two forms. Consider annotating the parameters in the two forms of equations to help illustrate the connections:

Explain to students that base 10 logarithms are often written without the subscript 10. For example, \(\log_{10} 1,\!000=3\) may be written as \(\log 1,\!000 = 3\), which is equivalent to \(10^3=1,\!000\). When no base is specified in a logarithm, it is understood that the base is 10. In all other logarithms, the base is shown.