Lesson Bone Density Math and Logarithm Introduction

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Quick Look

Grade Level: 11 (9-12)

Time Required: 1 hours 45 minutes

(two 50-minute sessions)

Lesson Dependency:

Subject Areas: Algebra

Graph of exponential and log functions.
Exponential and logarithm graph.
Copyright © 2006 Kristyn Shaffer, Vanderbilt University


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.

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.

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.

  • (+) Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents. (Grades 9 - 12) More Details

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  • Analyze how technology transfer occurs when a user applies an existing innovation developed for one function to a different purpose. (Grades 9 - 12) More Details

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  • (+) Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents. (Grades 9 - 12) More Details

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Worksheets and Attachments

Visit [www.teachengineering.org/lessons/view/van_bmd_less2] to print or download.

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Pre-Req Knowledge

Students need to have already covered exponential functions.


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.

A table provides the definition and example for four properties of logarithms: product, quotient, power and equality.
Copyright © 2006 Kristyn Shaffer, Vanderbilt University

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?


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.


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


Kristyn Shaffer; Megan Johnston

Supporting Program

VU Bioengineering RET Program, School of Engineering, Vanderbilt University


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: May 18, 2022

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