Copyright 2006 Pearson Education Inc Publishing as Pearson

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9 - 1

9 9. 1 9. 2 9. 3 9. 4 9. 5 9. 6 9. 7 Exponential and Logarithmic Functions Composite and Inverse Functions Exponential Functions Logarithmic Functions Properties of Logarithmic Functions Common and Natural Logarithms Solving Exponential and Logarithmic Equations Applications of Exponential and Logarithmic Functions Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

9. 5 n n Common and Natural Logarithms Common Logarithms on a Calculator The Base e and Natural Logarithms on a Calculator Changing Logarithmic Bases Graphs of Exponential and Logarithmic Functions, Base e Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Common Logarithms on a Calculator Here, and in most books, the abbreviation log, with no base written, is understood to mean logarithm base 10, or a common logarithm. Thus, log 21 = log 10 21. On most calculators, the key for common logarithms is marked LOG. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9 - 4

Example Use a scientific calculator to approximate each number to four decimal places. Solution a) We enter 2, 356 and press LOG. We find that Rounded to four decimal places Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9 - 5

Example Use a scientific calculator to approximate 102. 15 to four decimal places. Solution We enter 2. 15 and then press 10 x. On most graphing calculators, 10 x is pressed first, followed by 2. 15 and ENTER. Rounding to four decimal places we have Rounded to four decimal places Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9 - 6

The Number e Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9 - 7

The Base e and Natural Logarithms on a Calculator Logarithms base e are called natural logarithms, or Napierian logarithms, in honor of John Napier, who first “discovered” logarithms. The abbreviation “ln” is generally used with natural logarithms. Thus, ln 21 = loge 21. On most calculators, the key for natural logarithms is marked LN. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9 - 8

Example Use a scientific calculator to approximate ln 712 to four decimal places. Solution We enter 712 and press LN. We find that Rounded to four decimal places Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9 - 9

Example Use a scientific calculator to approximate e– 1. 33 to four decimal places. Solution We enter – 1. 33 and then press ex. On most graphing calculators, ex is pressed first, followed by – 1. 33 and ENTER. Rounding to four decimal places we have Rounded to four decimal places Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9 - 10

Changing Logarithmic Bases Most calculators can find both common logarithms and natural logarithms. To find a logarithm with some other base, a conversion formula is usually needed. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9 - 11

The Change-of-Base Formula For any logarithmic bases a and b, and any positive number M, (To find the log, base b, of M, we typically compute log M/ logb or ln M/ ln b. ) Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9 - 12

Example Find log 37 using the change-of-base formula. Solution Substituting into Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9 - 13

Graphs of Exponential and Logarithmic Functions, Base e Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9 - 14

Example Graph Solution x 0 1 – 1 2 – 2 y y 8 7 2 3. 7 1. 4 8. 4 1. 1 6 5 4 3 2 1 -5 -4 -3 -2 -1 1 2 -1 3 4 5 x -2 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9 - 15

Example Graph Solution x 0 1 – 1 2 – 2 y y 8 7 2 1. 4 3. 7 1. 1 8. 4 6 5 4 3 2 1 -5 -4 -3 -2 -1 1 2 -1 3 4 5 x -2 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9 - 16

Example Graph y = f (x) = ln(x – 1). y = ex + 1 y Solution y=x 6 x y 2 5 8 3/2 0 1. 4 1. 9 – 0. 7 5 4 y = ln(x– 1) 3 2 1 -3 -2 -1 -1 -2 1 2 3 4 5 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 6 7 8 9 x Slide 9 - 17