Lesson 18

In this warm-up, students practice evaluating and comparing logarithms. Except for one that requires estimation, all logarithms can be found mentally.

Students may need to rewrite some of the logarithms involving decimal numbers in order to better see the value of the logarithm. For example, they may find it helpful to write 0.2 as \(\frac 15\) before they could see that the value of the log with base 5 is -1. Similarly, they may think of \(\log 0.01\) as \(\log \frac{1}{100}\) or \(\log 10^{\text-2}\) to see that its value is -2. Monitor for students using this strategy to share during the whole-class discussion.

If needed, remind students that \(\ln\) is the logarithm for base \(e\).

Invite students to share their ordered list and their reasoning. A few points to highlight, if not already mentioned by students:

• We don’t need to know the value of \(e\) to evaluate \(\ln {1}\) because any base raised to an exponent of 0 results in 1.
• \(\log 11\) is greater than but close to \(\log 10\) (which is 1), so its value is greater than but close to 1.
• Sometimes it is helpful to rewrite a number for which we’re finding a logarithm in a different form, for example, writing 0.2 as \(\frac15\). If time allows, invite previously identified students to share how they rewrote the expressions to help make sense of the value of the logarithm.

This is the first of two optional activities that explore the pH scale as a real-world application of logarithmic functions. Students analyze a table of values showing different hydrogen ion concentrations and the corresponding pH ratings, and notice that the pH values are related to the exponents of the hydrogen ion concentrations. They then try to generalize the pattern with a description or an expression.

Tell students that they will look at how acidic different liquids are and how acidity is typically measured. Before explaining further and if practical, consider demonstrating the use of pH strips to test the acidity of a few liquids such as water, liquid soap, or salt water.

Then, explain to students that the pH scale is a way of measuring the acidity of different liquid solutions. Roughly speaking, it measures the concentration of positive hydrogen ions (\(H^+\)) in the solution. A small pH means that there are a lot of positive hydrogen ions and such a solution is called acidic. A large pH means that there are relatively few positive hydrogen ions and such a solution is called basic.

Instruct students to look over the list of liquids and ask, “Which drink has more hydrogen ions: coffee or water? How do you know?” Make sure students recognize, for example, that \(10^{\text-5}\) is greater than \(10^{\text-7}\).

Arrange students in groups of 2. Ask students to complete the first question (comparing the four drinks) and then pause for a brief class discussion. Once students see that orange juice is the most acidic drink because it has the greatest hydrogen ion concentration, provide students with the pH ratings of the drinks on the list and ask them to complete the rest of the task.

Focus the discussion on the last question. Invite students to share their descriptions or expressions. If not already mentioned in students’ explanations, highlight that each decrease of 1 in the pH rating represents an increase of hydrogen ion concentration by a factor of 10.

This is the second of two optional activities in which students explore pH ratings as an application of logarithmic functions. In the first activity, students noticed a pattern in the concentration of positive hydrogen ions to pH ratings and described it informally. In this activity, they represent the relationship with an equation, test it, and use it to solve problems.

Monitor for how students approach the third question. Some students may pay attention only to the exponent -11 in \(5.6 \times 10^{\text-11}\) and disregard the 5.6. Identify students who recognize that the pH scale is the base 10 logarithm of whatever number or expression represents the hydrogen ion concentration, not just the part expressed in the power of 10.

Ask students to look at the table in this activity and the one in the previous activity. Briefly discuss how they are alike and different, and whether the table still shows the same relationship between hydrogen ion concentration and pH, as in the previous task. Make sure students see that the relationship hasn’t changed and that the hydrogen ion concentrations can be written as powers of 10.

Students may struggle writing an equation that relates the hydrogen ion concentration to the pH. Refer them back to the previous activity where they noticed a relationship by writing the hydrogen ion concentration using an exponential expression rather than a decimal. 

Invite students to share how they constructed the equation in the first question. Ensure that they see pH rating as a negative base 10 logarithm function.

Focus the discussion on the third question. Help students see that the pH scale of milk of magnesia is \(\text- \log 5.6 \times 10^{\text-11}\). This log value can be computed with a calculator, but it can also be estimated. For example, we know that \(\text- \log 10^{\text-11}\) is 11 and \(\text- \log 10^{\text-10}\) is 10, and that a hydrogen concentration in the amount of the latter is 10 times that of the former. Because the hydrogen ion concentration of milk of magnesia is 5.6 times, not quite 10 times, \(10^{\text-11}\), its pH is between 10 and 11.

This activity allows students to apply exponential and logarithmic reasoning in the context of the Richter scale, a scale for measuring the intensity of earthquakes.

Explain to students that a scale called the Richter scale is used to report the magnitude of earthquakes. The scale was initially based on measurements taken by a writing instrument on a seismometer. The writing instrument records its physical displacement caused by an earthquake—the stronger the quake, the greater the displacement. The measurement is then converted into the Richter scale, taking into account the distance between the seismometer and the earthquake’s epicenter. (Modern seismometers function in a different way, and so they are able to record physical displacements that are too large to realistically measure with a classical seismometer.)

Arrange students in groups of 2. Display the table from the activity. Give students 1 minute of quiet think time and ask them to be prepared to share at least one thing they notice and one thing they wonder. After inviting students to share some things they noticed and wondered, help students make sense of the table by discussing questions such as:

• “What displacement, in meters, is recorded when an earthquake is given a Richter rating of 7?” (1 m)
• “What about 3?” (0.1 mm)
• “During an earthquake, a seismograph records a displacement of 10 mm. What is the Richter rating for that quake?” (5)
• “What about when the displacement is 0.01 mm?” (2)

If needed, ask students to pause for a class discussion after the second question so students can see more clearly the relationship between seismometer displacement and Richter rating before answering the last question.

If students struggle identifying a relationship between the displacement and the Richter rating of an earthquake, ask them to focus on the displacement exponents and the Richter ratings. How are these related?

Focus the discussion on the patterns in the table:

• The seismograph displacements are changing by a factor of 10, while the Richter ratings are increasing by 1.
• The logarithm of the displacements also increases by 1, moving to the right in the table: -6 for \(10^{(\text -6)}\), -5 for \(10^{(\text-5)}\) and so on.
• The logarithms of the displacements are all 7 less than the Richter rating \(R\), so this gives an equation \(R = 7 + \log_{10} d\) relating the displacement \(d\) and the Richter rating.

Invite students to test the equation, verifying that it produces the right Richter ratings for some powers of 10 in the table.