### Summary

Students continue an examination of logarithms in the Research and Revise stage by studying two types of logarithms—common logarithms and natural logarithm. In this study, 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 formula. Then students see how these types of logarithms can be applied to solve exponential equations. They compute a set of practice problems and apply the skills learned in class.

### Engineering Connection

### Educational Standards

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### Learning Objectives

- Define the number e.
- Define the common and natural logarithms.
- Use common and natural logarithms to evaluate expressions.
- Use the change of base formula to convert to a common or natural logarithm in order to evaluate expressions and solve equations.

### Introduction/Motivation

### Lesson Background and Concepts for Teachers

Common Logarithms

- A common logarithm has a base of 10.
- If no base is given explicitly, it is common.
- You can easily find common logs of powers of 10.
- You can use your calculator to evaluate common logs.

Natural Logarithms

- Natural logarithms have a base of e.
- We write natural logarithms as ln.
- In other words, log
_{e}x = ln x. - The mathematical constant e is the unique real number such that the derivative (the slope of the tangent line) of the function f(x) = e
^{x}is f '(x) = e^{x}, and its value at the point x = 0, is exactly 1. - The function e
^{x}so defined is called the exponential function. - The inverse of the exponential function is the natural logarithm, or logarithm with base e.
- The number
*e*is also commonly*defined*as the base of the natural logarithm (using an integral to define the latter), as the limit of a certain sequence, or as the sum of a certain series. - The number
*e*is one of the most important numbers in mathematics, alongside the additive and multiplicative identities 0 and 1, the constant*π*, and the imaginary number*i*. -
*e*is irrational, and as such its value cannot be given exactly as a finite or eventually repeating decimal. The numerical value of*e*truncated to 20 decimal places is: 2.71828 18284 59045 23536.

### Associated Activities

- Linear Regression of BMD Scanners - Students complete an exercise showing logarithmic relationships and how to find the linear regression of data that does not seem linear upon initial examination. They relate the number of BMD scanners to time.

### Attachments

- Common and Natural Logarithms (ppt)
- Common and Natural Logarithms (pdf)
- Common and Natural Logarithms with Examples (ppt)
- Common and Natural Logarithms with Examples (pdf)
- Common and Natural Logs: Student Notes (doc)
- Common and Natural Logs: Student Notes (pdf)
- Student Notes Answers (doc)
- Student Notes Answers (pdf)
- Common and Natural Logs: Practice Problems (doc)
- Common and Natural Logs: Practice Problems (pdf)
- Practice Problem Answers (doc)
- Practice Problem Answers (pdf)
- Solving Exponential Functions (ppt)
- Solving Exponential Functions (pdf)
- Solving Exponential Functions with Examples (ppt)
- Solving Exponential Functions with Examples (pdf)
- Solving Equations Using Logs: Practice (doc)
- Solving Equations Using Logs: Practice (pdf)
- Solving Equations Using Logs: Practice Answers (doc)
- Solving Equations Using Logs: Practice Answers (pdf)

### Assessment

*Practice Problems:*Assign students the practice problems. Grade their answers to assess the learning objectives.

### 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

Last modified: April 24, 2015