Unit 5, Lesson 18
Applications of Logarithmic Functions (Optional)
Algebra 2
18.1
Warm-up
Standards Alignment
Building On
Addressing
HSF-LE.A.4
For exponential models, express as a logarithm the solution to where , , and are numbers and the base is 2, 10, or ; evaluate the logarithm using technology.
Instructional Routines
NoneMaterials
NoneActivity Narrative
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 before they could see that the value of the log with base 5 is -1. Similarly, they may think of as or to see that its value is -2. Monitor for students who use this strategy.
Launch
If needed, remind students that is the logarithm for base . Select students who rewrite the given decimals into fractions or powers of the base, and ask them to share during the whole-class discussion.
Activity
NoneWithout using a calculator, put these expressions in order, from least to greatest. Be prepared to explain your reasoning.
Activity Synthesis
Invite students to share their ordered list and their reasoning. Include any selected students who rewrote decimal values, and ask them to share their reasoning. A few points to highlight, if not already mentioned by students:
- We don’t need to know the value of to evaluate because any base raised to an exponent of 0 results in 1.
- is greater than but close to (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 . 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
Activity
Optional
Instructional Routines
MLR6: Three ReadsMaterials
NoneActivity Narrative
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.
This activity uses the Three Reads math language routine to advance reading and representing as students make sense of what is happening in the text.
Launch
Arrange students in groups of 2.
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 () 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 is greater than .
Use Three Reads to support reading comprehension and sense-making about this problem. Display only the problem stem and the table, without revealing the instructions or questions.
- For the first read, read the problem aloud, and then ask, “What is this situation about?” (measuring the pH of different liquids) Listen for and clarify any questions about the context.
- After the second read, ask students to list any quantities that can be counted or measured. (the hydrogen ion concentration and the pH)
- After the third read, reveal the instructions to complete the table and the next question and ask, “What are some ways we might get started on this?” Invite students to name some possible starting points, referring to quantities from the second read. (look for a connection between the given columns of the table, then use the definition of acidity to figure out which drink is the most acidic)
18.3
Activity
Optional
Instructional Routines
MLR8: Discussion SupportsMaterials
NoneActivity Narrative
This is the second of two optional activities in which students explore pH ratings as an application of logarithmic functions. In this activity, students practice using the relationship between hydrogen ion concentration and pH.
Monitor for how students approach the first question. Some students may pay attention only to the exponent -11 in and disregard the 5.6. Identify students who recognize that the pH scale is the negative of the base-10 logarithm of the hydrogen ion concentration, not just the part expressed in the power of 10.
18.4
Activity
Optional
Instructional Routines
Notice and WonderMaterials
NoneActivity Narrative
This optional activity allows students to apply exponential and logarithmic reasoning in the context of the Richter scale, a scale for measuring the intensity of earthquakes. In addition to using logarithms, students must use unit conversions to reasonably estimate distances. To understand the situation mathematically, students must reason abstractly and quantitatively (MP2).
Launch
Arrange students in groups of 2.
Explain to students that a scale called the Richter scale is used to report the magnitude of earthquakes. The scale was initially read from a machine called a “seismometer” which included a writing instrument and paper. The writing instrument records how much it moves back and forth when affected by an earthquake.The movement of the writing instrument is called the “displacement” and the stronger the quake, the greater the displacement.
The displacement measurement is then converted into the Richter scale, taking into account the distance between the seismometer and the earthquake’s “epicenter” (or point on the surface of the earth directly above the main focus of an earthquake). 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, but they use the same relationship between the displacement caused by the earthquake and the Richter scale.
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?” ( meters, 0.0001 meters, or 0.1 millimeters)
“During an earthquake, a seismograph records a displacement of 10 millimeters. What is the Richter rating for that quake? Explain your reasoning.” (5. 10 millimeters is the same as 0.01 meters, or meters.)
“What about when the displacement is 0.01 mm? Explain your reasoning.” (2. This is 1,000 smaller than the answer for the previous question, so it is 3 multiples of 10 less. That means the Richter rating is 3 less, and .)
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.
Lesson Synthesis
The purpose of this discussion is for students to recognize that logarithmic functions can be useful in these contexts. Ask students,
- “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 have a very large range, using a different scale helps when comparing the values and to make more sense of the actual values.)
- “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.)
- “Some advanced mathematical work such as cryptography (writing and breaking codes) involves very large numbers such as . How could you figure out how many digits are in this number if it were written out?” (Take the logarithm, base 10, of the number, round it down, then add 1. For example,
- 99 has 2 digits and
- 10 has 2 digits and
- 123,456 has 6 digits and
- has 24,862,048 digits.)
Consider asking students to research 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
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, , in moles per liter. If the concentration is , then the acidity rating, or pH rating, is . For example, grapefruit juice has a hydrogen ion concentration of about mole per liter, so its acidity rating is about 3. The concentration of hydrogen ions in lemon juice is 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 , or simply .
When the exponent in a power of 10 increases by 1, say from to , the quantity changes by a factor of 10. This means that lemon juice has 10 times the hydrogen ion concentration of grapefruit juice and the pH is 1 less. Waterhas a pH rating of 7 which is 4 greater than the pH of grapefruit juice. Water’s pH of 7 means that it has mole of hydrogen ions per liter which is 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 meters, the Richter rating of the earthquake can be expressed as . We can check that when (a displacement of 1 micrometer), the Richter rating is 1. And when the displacement increases by a factor of 10, the exponent of increases by 1, so the Richter rating of the earthquake increases by 1.