### Summary

In their reading from activity 1 of this unit, students should have discovered the term "logarithm." It is at this point that they begin their study of logarithms. Specifically, students examine the definition, history and relationship to exponents; they rewrite exponents as logarithms and vice versa, evaluating expressions, solving for a missing piece. Students then study the properties of logarithms (multiplication/addition, division/subtraction, exponents). They complete a set of practice problems to apply the skills they have learned (rewriting logarithms and exponents, evaluating expressions, solving/examining equations for a missing variable.) Then they complete a short quiz covering what they have studied thus far concerning logarithms (problems similar to the practice problems). They consider how what they have learned moves them closer to answering the unit's challenge question.### Engineering Connection

Students learn the equation needed to calculate bone mineral density, which is a calculation that biomedical engineers make all the time. Through this lesson, students come to realize that they have not learned all the necessary math skills in order to calculate bone mineral density but gain the motivation to learn about logarithms.

### Pre-Req Knowledge

Students need to have already covered exponential functions.

### Learning Objectives

After this lesson, students should be able to:

- Define logarithms as exponents.
- Rewrite logarithms as exponents and vice versa.
- Apply the properties of logarithms (products, quotient, power and equality) to solve equations.

### More Curriculum Like This

**Common and Natural Logarithms and Solving Equations**

Students continue an examination of logarithms in the Research and Revise stage by studying two types of logarithms—common logarithms and natural logarithm. They take notes about the two special types of logarithms, why they are useful, and how to convert to these forms by using the change of base f...

**Bone Mineral Density Math and Beer's Law**

Students revisit the mathematics required to find bone mineral density, to which they were introduced in lesson 2 of this unit. They learn the equation to find intensity, Beer's law, and how to use it. Then they complete a sheet of practice problems that use the equation.

**Exploring Nondestructive Evaluation Methods**

Students learn about nondestructive testing, the use of the finite element method (systems of equations) and real-world impacts, and then conduct mini-activities to apply Maxwell’s equations, generate currents, create magnetic fields and solve a system of equations. They see the value of NDE and FEM...

###
Educational Standards
Each *TeachEngineering* lesson or activity is correlated to one or more K-12 science,
technology, engineering or math (STEM) educational standards.

All 100,000+ K-12 STEM standards covered in *TeachEngineering* are collected, maintained and packaged by the *Achievement Standards Network (ASN)*,
a project of *D2L* (www.achievementstandards.org).

In the ASN, standards are hierarchically structured: first by source; *e.g.*, by state; within source by type; *e.g.*, science or mathematics;
within type by subtype, then by grade, *etc*.

Each *TeachEngineering* lesson or activity is correlated to one or more K-12 science,
technology, engineering or math (STEM) educational standards.

All 100,000+ K-12 STEM standards covered in *TeachEngineering* are collected, maintained and packaged by the *Achievement Standards Network (ASN)*,
a project of *D2L* (www.achievementstandards.org).

In the ASN, standards are hierarchically structured: first by source; *e.g.*, by state; within source by type; *e.g.*, science or mathematics;
within type by subtype, then by grade, *etc*.

###### Common Core State Standards - Math

- (+) Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents. (Grades 9 - 12) Details... View more aligned curriculum... Do you agree with this alignment?

###### International Technology and Engineering Educators Association - Technology

- Technology transfer occurs when a new user applies an existing innovation developed for one purpose in a different function. (Grades 9 - 12) Details... View more aligned curriculum... Do you agree with this alignment?

###### Tennessee - Math

- (+) Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents. (Grades 9 - 12) Details... View more aligned curriculum... Do you agree with this alignment?

### Introduction/Motivation

Let's jump right into the equation that real-world engineers use to find bone mineral density. Do not be concerned about the many symbols and variables that you are unfamiliar with. I just want to give you idea of the larger picture of the unit. Once you see these equations, we will start in with logarithms by relating them to exponents, which is something that you are all familiar with.

### Lesson Background and Concepts for Teachers

On Day 1, present to the class the BMD Math and Logarithms Introduction, a PowerPoint® file. Before starting, hand out the History of Logarithms Handout and Exponential Functions Graphs Notes for students to take notes on while you go through the presentation content.

On Day 2, present the Properties of Logarithms, another PowerPoint® file. Then at lesson end or as homework, assign students to complete the Log Properties Practice Problems, a sheet of practice problems. Also administer the Log Properties Quiz.

Conclude with a class discussion: How might what you have learned in this lesson apply to the challenge question? How much closer are you to answering the questions? What do you still have to learn?

### Attachments

### Assessment

*Practice Problems*: At lesson end or as homework, assign students to complete the Log Properties Practice Problems. Review their answers to assess their depth of comprehension.

*Quiz*: Administer the eight-question multiple-choice Log Properties Quiz on logarithm properties. Review students' answers to gauge their comprehension.

### Contributors

Kristyn Shaffer; Megan Johnston### Copyright

© 2013 by Regents of the University of Colorado; original © 2006 Vanderbilt University### Supporting Program

VU Bioengineering RET Program, School of Engineering, Vanderbilt University### Acknowledgements

The contents of this digital library curriculum were developed under National Science Foundation RET grant nos. 0338092 and 0742871. However, these contents do not necessarily represent the policies of the NSF, and you should not assume endorsement by the federal government.

Last modified: November 6, 2017

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