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

2. Exponential and Logarithmic Functions 9.1Composite and Inverse Functions 9.2Exponential Functions 9.3Logarithmic Functions 9.4Properties of Logarithmic Functions 9.5Common and Natural Logarithms 9.6Solving Exponential and Logarithmic Equations 9.7Applications of Exponential and Logarithmic Functions 9

3. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 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 9.5

4. Slide 9- 4 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

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

6. Slide 9- 6 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Solution Use a scientific calculator to approximate 10 2.15 to four decimal places. Rounded to four decimal places We enter 2.15 and then press 10 x. On most graphing calculators, 10 x is pressed first, followed by 2.15 and. Rounding to four decimal places we have ENTER

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

8. Slide 9- 8 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 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 = log e 21. On most calculators, the key for natural logarithms is marked. LN

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

10. Slide 9- 10 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Solution Use a scientific calculator to approximate e –1.33 to four decimal places. Rounded to four decimal places We enter –1.33 and then press e x. On most graphing calculators, e x is pressed first, followed by –1.33 and. Rounding to four decimal places we have ENTER

11. Slide 9- 11 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 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.

12. Slide 9- 12 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 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.)

13. Slide 9- 13 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Solution Find log 3 7 using the change-of-base formula. Substituting into

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

15. Slide 9- 15 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Graph Solution xy 0 1 –1 2 –2 2 3.7 1.4 8.4 1.1

16. Slide 9- 16 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Graph Solution xy 0 1 –1 2 –2 2 1.4 3.7 1.1 8.4

17. Slide 9- 17 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Graph y = f (x) = ln(x – 1). Solution xy 2 5 8 3/2 0 1.4 1.9 –0.7