## 1.6 Exponential and Logarithm Functions

### Summary

The exponential and logarithmic functions are inverse functions of each other. Exploring this relationship between them, we discuss properties of the exponential and logarithm functions, including their graphs and the rules for manipulating exponents and logs. We define the important number e that is the base for the natural logarithm, and is the standard base that we use for exponential functions in calculus. The exponential function turns out to be very useful in certain kinds of population madeling.

By the end of your studying, you should know:

• The difference between an exponential growth rate and growth factor.
• The definition of a general exponential, including its graph.
• The definition of a general logarithm, including its graph.
• Laws of exponents.
• Laws of logarithms.
• Definition of the number e.
• The functions that comprise the Elementary Functions of calculus.

On-screen applet instructions: There is a show-and-hide button for the natural log function. How are ex and ln(x) related?

### Examples

The ground noise created by an airplane taking off at time t = 0 is measured in decibels, and is given by

where t is measured in seconds. How long will it be before the level of noise drops to 2 decibels?

Find the inverse of

and find the domains of f and f–1.

Find all solutions to the equation ln(x + 4) = 2ln(x) – ln(2).

### Applets

Comparing Exponential Functions

### Videos

See short videos of worked problems for this section.

### Exercises

See Exercises for 1.6 Exponential and Logarithm Functions (PDF).

Work online to solve the exercises for this section, or for any other section of the textbook.

### Resources on the Web

Information on Newton
Biographical data from St. Andrew's University's Web site
Excerpt from W.W. Rouse Ball's "A Short Account of the History of Mathematics"

Calculus Applications
Project Intermath

Exponentials and Logs
Wolfram Research

#### Interesting Application

World record times for the mile-- are they exponential?

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Copyright © 2005 Donald L. Kreider, C. Dwight Lahr, Susan J. Diesel