Logarithms Functions

Logarithms Functions Author: Hannah Knisely, Kent Island High School, Queen Anne’s County Public Schools Background Information Subject: Algebra II

Grade Band: 9-12

Duration: 90 Minutes Students will discover patterns within logarithmic functions, thus discovering logarithmic properties. They will analyze and apply logarithmic functions to sound frequencies and earthquakes. A STEM Overview: Specialist will help students interpret their findings with the properties of logarithms as well as engage students in hands-on learning experiences that allow them to draw connections between course content ant the work performed by STEM professionals. Background Information: Teachers should familiarize themselves with the Properties of Logarithms:        

Additionally, teachers should understand the scales for both decibels and magnitude and how they relate to logarithms. dB 1 10 20 30 40 50 60 70 80 90 100 Sound 100 101 102 103 104 105 106 107 108 109 1010 Intensity

The Richter scale was developed to assign a single number to quantify the energy released during an earthquake. The scale is a base-10 logarithmic scale. The magnitude is defined as the logarithm of the ratio of the amplitude of waves measured by a seismograph to an arbitrary small amplitude. An earthquake that measures 5.0 on the Richter scale has a shaking amplitude 10 times larger than one that measures 4.0, and corresponds to a 31.6 Logarithms Functions Background Information times larger release of energy. The STEM Specialist can:  Help students decipher patterns between logarithmic expressions. STEM Specialist Connection:  Engage students in hands-on learning experiences on the application of logarithms in the workplace. STEM Specialists can be found at www.theSTEMnet.com  Logarithms allow us to solve for exponential equations in which the variable is in the exponent.  Logarithmic functions can be used to model seismic activity, sound intensity, magnitude of Enduring Understanding: earthquakes, compute compound interest in terms of time, calculate population growth, radioactive decay, etc.  Logarithms were developed to multiply large numbers without the use of a calculator.

1. How can logarithms be used to solve exponential functions in which the variable is found in Essential Questions: the exponent? 2. How do STEM professionals apply logarithmic functions to develop solutions to problems?

Students will be able to: 1. Convert exponential functions to logarithmic functions and vice versa. 2. Solve base 10 logarithms. Student Outcomes: 3. Apply the properties of logarithms to simplify and expand different logarithmic functions. 4. Interpret and solve real world situations in which logarithms would be necessary to solve the problem. Audience: ☒Peers ☒Experts / Students will work independently to derive the properties of logarithmic Practitioners Product, Process, Action, Performance, etc.: functions and apply them to real-world problems. ☒Teacher(s) ☐School Community ☐Other______Logarithms Functions Background Information Domain: Modeling with Functions Cluster: Construct and compare linear, quadratic, and exponential models and solve problems Standard: F.LE.A.4 For exponential models, express as a logarithm the solution to abct=d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the Standards Addressed in the Unit: logarithm using technology.

Cluster: Build a function that models a relationship between two quantities. Standard: F.BF.A.1: Write a function that describes a relationship between two quantities. Equipment:  computer with internet access  projector  Scientific or Graphing calculator  Computers for each student (optional)  Earphones (optional) Website* http://www.dangerousdecibels.org/virtualexhibit/

* Students are linked to an online site for the engagement activity. The site has been chosen for its content and grade-level appropriateness. Teachers should preview website before introducing the activities to students and adhere to their school system’s policy for internet use. Suggested Materials and Resources: People, Facilities:  STEM Specialist  Computer lab (optional) Materials (rubrics, worksheets, PowerPoints, answer keys, etc.):  Logarithms PowerPoint  Discovering Properties of Logarithms  Homework  Homework (Answers)  Calculator  Computer (with internet access)  Projector Logarithms Functions Logarithms Functions

Learning Experience 5E Component Standards for Mathematical Identify the 5E component addressed for the Details Practice learning experience. The 5E model is not linear. Materials: ☒Engagement ☐Make sense of problems and  Computer with internet access persevere in solving them.  Projector ☐Exploration  Logarithms PowerPoint ☐Reason abstractly and ☐Explanation  Individual Computers for students (optional) quantitatively.  Earphones (optional) ☐Extension ☐Construct viable arguments Preparation: and critique the reasoning of ☐Evaluation  Explore others. http://www.dangerousdecibels.org/virtualexhibit/ on your own. Understand how the website works and ☐Model with mathematics. make a decision whether you want your students to explore the site individually or as a class. ☒Use appropriate tools  Familiarize yourself with decibels and sound intensity. strategically. Be able to explain the different scales to your students. ☐Attend to precision.  Use PowerPoint slides 1-4 as a guide through the engagement portion of the lesson. ☐Look for and make use of structure. Facilitation of Learning Experience:  Begin your lesson with the Dangerous Decibels Activity ☐Look for and express (http://www.dangerousdecibels.org/virtualexhibit/). regularity in repeated reasoning. Have students—individually or as a class—explore the website. You can choose to do as many (or as little) activities as your class time allows. Be sure not to leave out the “measuring sound” section as that is the piece that explains the decibel scale. Students should be carefully monitored during this activity since the sounds can be pretty loud. Logarithms Functions Learning Experience 5E Component Standards for Mathematical Identify the 5E component addressed for the Details Practice learning experience. The 5E model is not linear.  If you choose to allow the students to explore the website on their own, you will need to monitor their use of time in each section.  As a class, discuss the results from each section of the Dangerous Decibels activity. If you are completing the entire activity as a class this should be done throughout. If the students are doing this activity individually, you will need to discuss each element of their exploration. Compare and contrast how each student did with the activity.  Continue your discussion on decibel levels by showing the next 3 graphics on the PowerPoint—infographic (bubbles), lesser water boatman graph of noise levels, and the frequencies heard by different animals.

Materials: ☐Engagement ☐Make sense of problems and  Computer persevere in solving them.  Projector ☐Exploration  Logarithms PowerPoint ☐Reason abstractly and ☒Explanation quantitatively. Preparation: Use PowerPoint slides 5-11 as a guide through the ☐Extension ☐Construct viable arguments explanation section. and critique the reasoning of ☐Evaluation others. Facilitation of Learning Experience:  Discuss the origin of logarithms—where they come ☐Model with mathematics. from, what they were (still are) used for, etc.  Be sure to indicate how you switch from logarithmic to ☐Use appropriate tools exponential form and vice versa. The equation for this strategically. process is found at the bottom of slide 8. Continue this Logarithms Functions Learning Experience 5E Component Standards for Mathematical Identify the 5E component addressed for the Details Practice learning experience. The 5E model is not linear. discussion by working through a few examples with ☐Attend to precision. your students (found on the next two slides).  Next, discuss the common bases used in everyday life ☒Look for and make use of (Base 10, 2, and e) structure.  Ask the students when they have used logarithms (such as in chemistry class) and why logarithms were ☒ Look for and express used. regularity in repeated reasoning.

Materials: ☐Engagement ☒Make sense of problems and  Discovering Properties of Logarithms worksheet persevere in solving them.  Pencil/Pen ☒Exploration  Calculator ☒Reason abstractly and ☐Explanation  Computer quantitatively.  Projector ☐Extension  Logarithms PowerPoint ☒Construct viable arguments and critique the reasoning of ☐Evaluation Preparation: others.  Use PowerPoint slides 12-17 as a guide through the exploration section. ☐Model with mathematics.  Complete the Discovering Properties of Logarithms worksheet in advance to familiarize yourself with the ☒Use appropriate tools patterns the students should discover. strategically.

Facilitation of Learning Experience: ☒Attend to precision.  Provide each student with a copy of discovering properties of logarithms worksheet. Allow students to ☒Look for and make use of work in partners to complete the assignment. structure. Students should notice the patterns that develop in the first column between addition and multiplication. ☐ Look for and express They should also notice the pattern between regularity in repeated reasoning. Logarithms Functions Learning Experience 5E Component Standards for Mathematical Identify the 5E component addressed for the Details Practice learning experience. The 5E model is not linear. subtraction and division in the second column. Finally, they should notice that exponents are directly related to multiplying by a coefficient of the same numerical value.  Discuss as a class the discoveries that were made among the pairs in the class.  Continue that discussion with slides 12 - 15 of the PowerPoint. The properties the students should have discovered are present here. There are a few other properties that should be noted in the student’s notes that were not discovered with the activity. Be sure they are included in their notes as shown on slide 15.  Use these properties to work through the next two examples as a class. These examples are worked through on slides 16 & 17.

Materials: ☐Engagement ☒Make sense of problems and  Logarithms PowerPoint persevere in solving them.  Computer ☐Exploration  Projector ☒Reason abstractly and ☐Explanation quantitatively. Preparation: Use slides 18 – 23 as a guide through the extension ☒Extension ☒Construct viable arguments section of the lesson. and critique the reasoning of ☐Evaluation others. Facilitation of Learning Experience:  As a class, work through the next two examples (slides ☐Model with mathematics. 19-23) together. The first example revisits the idea of decibels and asks the students to determine just how ☒Use appropriate tools much louder one decibel level is than another in terms Logarithms Functions Learning Experience 5E Component Standards for Mathematical Identify the 5E component addressed for the Details Practice learning experience. The 5E model is not linear. of relative intensity. strategically.  Review the earthquake graph with your students. Point out different magnitude levels and show them ☒Attend to precision. the comparable energy equivalent. Many will be surprised by the amount of energy emitted from an ☐Look for and make use of earthquake and the comparable events. structure.  The earthquake example compares two different earthquakes magnitudes and asks the students to ☒ Look for and express solve just how much more energy one earthquake had regularity in repeated reasoning. than the other. Materials: ☐Engagement ☒Make sense of problems and  Logarithmic Functions Homework persevere in solving them.  Logarithmic Functions Answer Key ☐Exploration  Pencil/Pen ☒Reason abstractly and ☐Explanation  Calculator quantitatively.

☐Extension Preparation: ☐Construct viable arguments Contact the STEM Specialist in advance to co-plan the and critique the reasoning of ☒Evaluation learning experience for students. STEM Specialist can be others. found at wwww.theSTEMnet.com. A description of the ability level of the students, as well as some of the prior ☒Model with mathematics. knowledge your students may have of logarithmic functions may be helpful to the STEM Specialist prior to ☒Use appropriate tools the visit. strategically.

Facilitation of Learning Experience: ☒Attend to precision.

 Have the STEM Specialist engage students in the ☐Look for and make use of predetermined learning experience emphasizing how structure. logarithmic functions are used in the workplace. Logarithms Functions Learning Experience 5E Component Standards for Mathematical Identify the 5E component addressed for the Details Practice learning experience. The 5E model is not linear. ☒ Look for and express  After the STEM Specialist has engaged students in the regularity in repeated reasoning. learning experience, provide each student a copy of the homework assignment to work on independently after class. At the start of the next class, you can review answers or collect papers for grading.

Supporting Information Interventions/Enrichments Struggling Learners Identify interventions and enrichments for diverse learners. Students should be provided with copies of notes or a note guide to follow along with the PowerPoint. Additionally, students could work with the teacher or STEM specialist in the discovery activity to better guide them to the patterns present. A list of the different properties of logarithms can (should) be provide to struggling learners to help aide them in solving difficult equations.

English Language Learners

Similarly to struggling learners accommodations should be provided. A vocabulary list may be helpful to ELL students (logarithm, exponent, magnitude, decibel, etc.) Many words in this lesson are comparable to European languages because of the origin of the words therefore, vocabulary may be helpful but is not necessarily needed in all ELL students. Logarithms Functions Gifted and Talented

Expand student knowledge by providing more complex, challenging questions. Begin with the basics that all students are learning. Give them more advanced questions as practice problems if they should finish early. Match these students with some of the struggling learners for the discovery lesson to aide those students as well as enhance the understanding of your gifted and talented students. As an additional extension, have the gifted and talented students research magnitudes of different earthquakes around the world and write their own example problems (providing the answers and work along with it).

Investigation Part One Investigation Part Two Investigation Part Three

Use your calculator to find the Use your calculator to find the Use your calculator to find the following common logs: following common logs: following common logs:                                 Write down any patterns or connections you recognize between your answers. What relationships can you identify?

Develop a general rule to write as a Develop a general rule to write as a Develop a general rule to write as the single logarithm, for all positive values single logarithm, for all positive values logarithm of a single constant, for all Logarithms Functions of A and B. of A and B. positive values of A and all real values of n.

Test your general rule on several other values for A and B. Test your general rule on several other values for A and n. Logarithmic Functions Homework Write the following equations in logarithmic/exponential form

Solve each equation. 5. 9.

6. 10.

7. 11.

8. 12. An equation for loudness, in decibels, is , where R is the relative intensity of the sound. Sounds that reach levels of 120 decibels or more are painful to humans. What is the relative intensity of 120 decibels?

13. 14. 15. 16. 17. 18. An earthquake rated at 3.5 on the Richter scale is felt by many people, and an earthquake rated at 4.5 may cause local damage. The Richter scale magnitude reading is given by , where represents the amplitude of the seismic wave causing ground motion. How many times greater is the amplitude of an earthquake that measures 4.5 on the Richter scale than on the measures 3.5?

19. 20. 21. 22. 23. 24. 25. Rob is solving a problem involving logarithms. He does everything correctly except for one thing. He mistakenly writes . However, after substituting the values for and in his problem, he amazingly still gets the right answer! The value of was 11. What must the value of have been?

26. 27. 28. Logarithmic Functions Homework Write the following equations in logarithmic/exponential form

1. 53 = 125 1 3. 77765 = 6

2. 𝑙𝑜𝑔 0.00001 = −5 3 10 4. 𝑙𝑜𝑔 8 = 32 5

Solve each equation.

3 5. 𝑙𝑜𝑔𝑥= 9. 𝑙𝑜𝑔45 + 𝑙𝑜𝑔4𝑥= 𝑙𝑜𝑔460 4 2

10. 𝑙𝑜𝑔𝑞− 𝑙𝑜𝑔3 = 𝑙𝑜𝑔7 6. 𝑙𝑜𝑔127 = 𝑥 2 2 2 3

11. 𝑙𝑜𝑔2ሺ𝑥+ 4ሻ− 𝑙𝑜𝑔2ሺ𝑥− 3ሻ = 3 7. 𝑙𝑜𝑔𝑏1024 = 5

12. 𝑙𝑜𝑔510 + 𝑙𝑜𝑔512 = 3𝑙𝑜𝑔52 + 𝑙𝑜𝑔5𝑎 8. 𝑙𝑜𝑔7ሺ𝑥+ 6ሻ = 𝑙𝑜𝑔7ሺ8𝑥+ 20ሻ

13. An equation for loudness, in decibels, is 𝐿= 10𝑙𝑜𝑔10𝑅, where R is the relative intensity of the sound. Sounds that reach levels of 120 decibels or more are painful to humans. What is the relative intensity of 120 decibels?

30. 14. An earthquake rated at 3.5 on the Richter scale is felt by many people, and an earthquake rated at 4.5 may cause local damage. The Richter scale magnitude reading is given by , where represents the amplitude of the seismic wave causing ground motion. How many times greater is the amplitude of an earthquake that measures 4.5 on the Richter scale than on the measures 3.5?

31. 32. 33. 34. 35. 36. 15. Rob is solving a problem involving logarithms. He does everything correctly except for one thing. He mistakenly writes . However, after substituting the values for and in his problem, he amazingly still gets the right answer! The value of was 11. What must the value of have been?

37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47.