Lesson and Assessments - Introduction to Logarithms

Progression of Skills (outline)

Before the lesson:

• Students should have an understanding of inverse functions and exponential functions. 

During this toolkit lesson:

• Students should focus on defining logarithms by utilizing exponential functions as motivation, graphs of logarithms and solving exponential equations.

After the lesson:

• Students should utilize their knowledge of logarithms and exponentials to create equations and solve real-world problems

Lesson Activity 1:

Ladder of Logs (Answer Key) ~ approx. 40 minutes

• Each student needs a handout and a calculator. Students can work individually or in small groups. 

Directions for Intervention Activity:

1. Students can work individually or in small groups. If students are working individually, encourage them to discuss answers with others as they progress through the activity. Ask students to share their thoughts with the whole class. If students are working in small groups, they should work together on a common space (whiteboard or piece of paper). 
2. For the chess board problem:
   1. Make sure students understand that the numbers inside the chess board are the labels.
   2. Have students discuss their answers to part d. 
   3. Ask students, “What is different about your strategy and your partners’?” or “Are your strategies the most efficient?” or “What are some limitations of your strategies?”
   4. Students may raise 2 to the nth power instead of the (n-1) power. 
3. For the Successive Splitting problem:
   1. Students may need reminded of the equation for exponential growth: f(x)=(initial population)(base)^x
   2. If students just write 34 instead of 100•34, ask them if their answer makes sense in the context of the problem. 
   3. Another area that may provide a struggle for students is making sure to switch from minutes to hours in part b. 
4. In the interest problem, students may need to be reminded of the interest formula:    Total =(initial amount)(1+r)t, where r is the interest rate (write % as decimal) and t is the time the initial amount is earning that interest
5. In the Reflect and Brainstorm section, you can ask guiding questions to help students realize that all of the functions were exponential and we are looking for the exponent.  

Progress Monitoring:

• Make sure students are writing the exponentials correctly. Listen to their explanations of how they think they would solve for the exponents. Additionally, listen to their language that they are using the mathematically correct terms, such as base and power.

Anticipate Misconceptions:

• Logarithm structure is similar to exponential structure. Students may put the power where the value should be. For instance, if b^p=n and log_bn=p, students may write log_bp=n.

Lesson Activity 2:

Exploring the Forest of Logs ~ approx. 40 minutes

• Each student will need a copy of  Exploring the Forest of Logs and a calculator. Students can work individually or in small groups. (Answer Key)

Directions for Intervention Activity:

1. Students may struggle with understanding the table. Allow pairs to share their thoughts with the group, but some productive struggle is good here. 
2. On page 2, problem three is attempting to have students understand how different bases impact the variable. 
3. On the fourth page, have students verbalize saying the logarithm. There are some notes about the common logarithm and natural logarithm there for completeness. You can talk about different bases here as well. 
4. To practice the new definition of logarithm, revisit the Ladder of Logs Activity and write all of the exponential equations and expressions as logarithms. 
   1. Students can also practice inputting logarithms in the calculator as they write the logarithmic function. 

Progress Monitoring:

• Make sure students are following the structure of logarithms and rewriting the exponential functions correctly.

Lesson Activity 3:

Math Medic Graphs of Logs Activity ~ approx. 40 minutes

• Each student will need a copy of Graphs of Logs by Math Medic and a calculator.
• Students can work individually to apply some of their knowledge from the previous activity to evaluate and complete the tables. Students can discuss with a neighbor to check their answers and discuss strategies. (Answer Key)

Directions for Intervention Activity:

1. Students could try to complete this by hand first and then use a graphing calculator to check their answer. 
2. Students may need to be reminded of the prerequisite skills of y-intercept, domain, range, and asymptote.
3. Ask guiding questions to get them to see that f(x) and g(x) are inverses. 
   1. You can discuss the box with y-intercept, domain, range, and asymptote as well as the horizontal line test to see if f(x) is invertible.
4. Ask students how they think this relationship may be useful.

Progress Monitoring:

• Check student’s graphs and tables to make sure they are understanding logarithms properly.

• Richter Scale: Why is a magnitude 7 earthquake so much more destructive than a magnitude 6? Is it just 'one more' on the scale?
• Sound Intensity (Decibels): How many times louder is a rock concert (120 dB) than a normal conversation (60 dB)? Why do we use decibels instead of just measuring sound directly?
• pH: If lemon juice has a pH of 2 and vinegar has a pH of 3, which is actually more acidic, and by how much?
• Half-life/Doubling Time: If a population of bacteria doubles every hour, why can't we just multiply by 2 repeatedly to find how many we'll have after 24 hours?
• Population Growth: In 1960, Earth had 3 billion people. By 2000, we hit 6 billion. If this pattern continued, would we expect 12 billion people by 2040? What's really happening with population growth?
• Scoville Units: If a jalapeño is 5,000 Scoville units and a ghost pepper is 1,000,000 Scoville units, how many jalapeños would you need to equal the heat of one ghost pepper?

• Díaz-Berrios, T., & Martínez-Planell, R. (2022). High school student understanding of exponential and logarithmic functions. Journal of Mathematical Behavior, 66, N.PAG. https://doi-org.ezproxy2.library.colostate.edu/10.1016/j.jmathb.2022.100953 
• Douventzidis, A., & Landquist, E. (2021). Logarithms Are Hot Stuff: A New Rating Scale for Chili Peppers. PRIMUS, 32(5), 650–660. https://doi-org.ezproxy2.library.colostate.edu/10.1080/10511970.2021.1886206