# Lesson 18

Applications of Logarithmic Functions

## 18.1: Scrambled Logs (5 minutes)

### Warm-up

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.

### Launch

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

### Student Facing

Without using a calculator, put the following expressions in order, from least to greatest. Be prepared to explain your reasoning.

• $$\log 11$$
• $$\log_2 8$$
• $$\log_5 0.2$$
• $$\log 0.01$$
• $$\ln 1$$

### Activity Synthesis

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.

## 18.2: How Acidic Is It? (15 minutes)

### Optional activity

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.

### Launch

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.

liquids hydrogen ion concentration
(moles per liter)
pH
water $$10^{\text-7}$$ 7
coffee $$10^{\text-5}$$ 5
root beer $$10^{\text-4}$$ 4
orange juice $$10^{\text-3}$$ 3
Design Principle: Support sense-making
Representation: Internalize Comprehension. Activate or supply background knowledge. Provide students with access to an image of a pH scale with ions. Select an image that shows the cluster and amount of ions (typically as a mass of circles), and clearly displays the correlation between the amount and where it falls on the pH scale. Allow students to take a moment to notice these features before moving into the questions. While students work, encourage them to draw upon their observations to support applied conceptual understanding. (To ensure that students make their own calculations, it may not be desirable to keep this image displayed throughout the lesson if the image contains numerical calculations.)
Supports accessibility for: Visual-spatial processing; Conceptual processing

### Student Facing

The pH scale is a way to measure the acidity of a liquid solution. It is based on the concentration of positive hydrogen ions in the liquid. A smaller pH indicates more hydrogen ions and higher acidity. A larger pH indicates less hydrogen ions and lower acidity.

Here is a table showing the hydrogen ion concentration (in moles per liter) and the pH of some different liquids:

liquids hydrogen ion concentration
(moles per liter)
pH
water $$10^{\text-7}$$ 7
coffee $$10^{\text-5}$$
root beer $$10^{\text-4}$$
orange juice $$10^{\text-3}$$
seawater
vinegar
1. Which of the drinks listed, water, coffee, root beer, or orange juice, is the most acidic? Which is the least acidic? Explain how you know.
1. Seawater has a pH of 8. Is it more acidic or less acidic than water? Record the hydrogen ion concentration of seawater in the table.
2. Vinegar has a pH of 2.4. Is it more acidic or less acidic than orange juice? Record the hydrogen ion concentration of vinegar in the table.
2. A logarithm is used to translate hydrogen ion concentrations to pH values. With a partner, discuss how the hydrogen ion concentrations might be related to the pH. Use words or expressions to describe the relationship you notice.

### Activity Synthesis

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.

## 18.3: pH Ratings (15 minutes)

### Optional activity

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.

### Launch

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.

Representation: Internalize Comprehension. Use color coding and annotations to highlight connections between representations in a problem. For example, invite students to write the scientific notation form of the number next to the number itself on the table. Students may choose to color code or draw arrows to brainstorm their interpretation of the connection between this column and the pH output. For example, students may draw an arrow from the exponent to the pH and write “opposite.”
Supports accessibility for: Visual-spatial processing

### Student Facing

This table shows the relationship between hydrogen ion concentrations and pH ratings (acidity) for different substances.

substance hydrogen ion concentration
(moles per liter)
pH
mild detergent 0.0000000001 10
toothpaste 0.000000001 9
baking soda 0.00000001 8
blood 0.0000001 7
milk 0.000001 6
banana 0.00001 5
tomato 0.0001 4
apple 0.001 3
lemon 0.01 2
1. Write an equation to represent the pH rating, $$p$$, in terms of the hydrogen ion concentration $$h$$, in moles per liter.
2. Test your equation by using the hydrogen ion concentration of a substance from the table as the input. Does it produce the right pH rating as the output? If not, revise your equation and test it again.
3. Magnesium hydroxide (also called “milk of magnesia”) is a medication used to treat stomach indigestion. It has a hydrogen concentration $$5.6 \times 10^{\text-11}$$ mole per liter. Estimate a pH rating for magnesium hydroxide. Explain or show your reasoning.
4. As shown in the table, apple has a pH of 3 and milk has a pH of 6. How many times more acidic is the apple than milk?

### Student Facing

#### Are you ready for more?

The graph shows points representing the hydrogen ion concentration, in moles per liter, and pH ratings of the different substances you saw earlier.

1. Which point represents baking soda? Which represents banana? How can you tell?
2. Vinegar has a pH of 2.4. Where on the graph would a point that represents vinegar be plotted?
3. Why do you think the graph appears the way it does, with a group of points stacked up along the vertical axis?
4. How is it like and unlike other graphs of logarithmic functions you have seen so far?

### Anticipated Misconceptions

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.

### Activity Synthesis

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.

Speaking, Representing: MLR8 Discussion Supports. Use this routine to support whole-class discussion. After each student shares their response to the first question, provide the class with the following sentence frames to help them respond: "I agree because . . .” or "I disagree because . . . .” If necessary, revoice student ideas to demonstrate mathematical language use by restating a statement as a question in order to clarify, apply appropriate language, and involve more students. For example, a statement such as, “The equation is a log” can be restated as a question such as, “Does the equation showing the relationship between hydrogen ion concentrations and pH ratings represent a logarithmic function?” This will help students solidify their understanding that pH rating is a negative base 10 logarithm function.
Design Principle(s): Support sense-making

## 18.4: Measuring Earthquake Strength (15 minutes)

### Optional activity

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.

### Launch

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.

Representation: Internalize Comprehension. Activate or supply background knowledge. Cut the table horizontally so the Richter rating and the seismograph displacement rows are able to be separated and manipulated independently of one another. While lining them up, slide the Richter rating row so the 1 in the Richter rating row is directly under the $$10^1$$ in the seismograph displacement row. Ask students to come up with the equation if the values were like this hypothetical arrangement. Once students arrive at the relationship of $$R=log_{10}d$$, then slide the values back into alignment and ask students what operation do they need to arrive at the accurate equation for the real table (add 7, resulting in $$R=log_{10}d+7$$).
Supports accessibility for: Memory; Conceptual processing

### Student Facing

Here is a table showing the Richter ratings for displacements recorded by a seismograph 100 km from the epicenter of an earthquake.

 seismograph displacement (meters) Richter rating $$10^{\text-6}$$ $$10^{\text-5}$$ $$10^{\text-4}$$ $$10^{\text-3}$$ $$10^{\text-2}$$ $$10^{\text-1}$$ $$10^0$$ $$10^1$$ 1 2 3 4 5 6 7 8
1. Compare an earthquake rated with a magnitude of 5 on the Richter scale and that rated with a 6. How do their displacements compare? What about an earthquake with a magnitude rated with a 2 and that rated with a 3?

2. Discuss with a partner how the displacement might be related to the Richter scale. Express that relationship in words or with an expression.
3. An earthquake shook the northwest part of Indonesia in 2004, causing massive damage and casualties. If a seismograph was located 100 km from the epicenter, it would have recorded a displacement of 125 m! Use your answer to the previous question to estimate the Richter rating for the earthquake.

### Anticipated Misconceptions

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?

### Activity Synthesis

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.

## Lesson Synthesis

### Lesson Synthesis

In this lesson, we saw logarithmic functions used to report some measurements. Encourage students to reflect on how and why logarithmic functions came in handy in these contexts.

• “Why not use the measured quantities to report, say, acidity and earthquake strengths? In other words, why use another scale rather than the actual measurements?” (When the numbers used to measure the quantities are either very large or very small, or both, using a different scale helps show them together and make more sense of them.)
• “What kinds of quantities can be better understood if expressed in terms of logarithmic functions?” (Quantities where there is a lot of variation in magnitude, like acidity.)
• “For what kinds of quantities are log functions not needed or helpful?” (Quantities where there is not too much variation, for example, the speed of cars, height of people, or distance between countries.)

Consider sharing some other measurements that are reported after being transformed with log functions like intensity of sound, intensity or exposure of light, the brightness of stars, and relative pitches of notes in music.

## Student Lesson Summary

### Student Facing

Logarithms are helpful in a variety of real-world contexts. Let’s look at an example in chemistry.

The acidity of a substance is measured by the concentration of positive hydrogen ions, $$H$$, in moles per liter. If the concentration is $$10^{x}$$, then the acidity rating, or pH rating, is $$\text{-}x$$. For example, grapefruit juice has a hydrogen ion concentration of about $$10^\text{-3}$$ mole per liter, so its acidity rating is about 3. The concentration of hydrogen ions in lemon juice is $$10^\text{-2}$$ mole per liter, so its acidity or pH rating is 2.

We can see that the pH rating is -1 times the exponent in the expression representing the hydrogen ion concentration. Because the exponent in a power of 10 can be expressed in terms of the base 10 logarithm, the pH rating can be expressed as $$\text-1 \log_{10} H$$ or simply $$\text- \log_{10} H$$.

When the exponent in a power of 10 increases by 1, say from $$10^\text{-3}$$ to $$10^\text{-2}$$, the quantity changes by a factor of 10. This means that lemon juice has 10 times the hydrogen ion concentration of grapefruit juice. Water, which has a pH rating of 7, has $$10^\text{-7}$$ mole of hydrogen ions per liter. This means that water has $$\frac{1}{10,000}$$ of the hydrogen ion concentration of grapefruit juice.

Another example of logarithm use is the Richter scale, which measures the strength of an earthquake in terms of the displacement of the needle on a seismograph. A displacement of 1 micrometer, one millionth of a meter, measures 1 on the Richter scale. Each time the displacement increases by a factor of 10, the Richter scale measure increases by 1. So a displacement of 10 micrometers measures 2 on the Ricther scale, and a displacement of 1,000 micrometers (1 mm) measures 4 on the Richter scale.

If the seismograph displacement is $$d$$ meters, the Richter rating of the earthquake can be expressed as $$7 +\log_{10}{d}$$. We can check that when $$d = 1 \times 10^{\text-6}$$ (a displacement of 1 micrometer), the Richter rating is 1. And when the displacement increases by a factor of 10, the exponent of $$d$$ increases by 1, so the Richter rating of the earthquake increases by 1.