1. L OGARITHMIC F UNCTIONS Section 8.4

2. 8.4 L OGARITHMIC F UNCTIONS Objectives: 1.Write logarithmic functions in exponential form and back. 2.Evaluate logs with and without a calculator. 3.Evaluate logarithmic functions. 4.Understand logs and inverses. 5.Graph logarithmic functions. Vocabulary: logarithm, common logarithm, natural logarithm

3. In Section 8.3, we learned that if the interest of a bank account is 5% compounded, then the total asset after t years is described by: Yearly:A t = P (1 + 0.05 ) t Monthly:A t = P (1 + 0.05 / 12) 12·t Daily:A t = P (1 + 0.05 / 365) 365·t Continuously:A t = P e 0.05·t

4. In each case, as long as we know the time, t, we can calculate the final (total) asset: Yearly:A 5 = P (1 + 0.05 ) 5 Monthly:A 10 = P (1 + 0.05 / 12) 12·10 Daily:A 2 = P (1 + 0.05 / 365) 365·2 Continuously:A 6 = P e 0.05·6

5. Now we would like to ask a reverse question: How long does the initial deposit (investment) take to reach the target asset value? Yearly:2000 = 1200 (1 + 0.05 ) t L ET ’ S T HINK

7. O S W E G O I NTRODUCING … O S W E G O hichhich xponentxponent oesoes n

8. E VALUATE THE E XPRESSIONS Think: “Which exponent goes on 2 to give me 8?” 3 2 3 0 Sorry, but “wego” does not really exist! In math, we use “logarithms.” The problems above would be written with the word “log” instead of “wego.”

9. E VALUATE THE E XPRESSIONS 4 2 -2 -3 6 Which Exponent Goes On…

10. S PECIAL L OGARITHM V ALUES

11. Definition: Logarithm of y with base b Let b and y be positive numbers, and b ≠ 1. Then, log b y = x if and only if y = b x. Definition: Exponential Function The function is of the form: f(x) = a · b x, where a ≠ 0, b > 0 and b ≠ 1. R EMEMBER THIS …?

12. R EWRITING L OGARITHMIC E QUATIONS

13. C OMMON N OTATION

14. E VALUATING C OMMON & N ATURAL L OGS

15. Examples: Evaluate the common and natural logarithms. a)log4 b)ln(1/5) c) lne -3 d)log(1/1000) 0.602 -1.609 1 -3

16. Practice: Evaluate the common and natural logarithm. a)ln0.25 b) log3.8 c)ln3 d) lne 2007 0.845 -1.386 0.580 1

17. pg. 490 #16-19, 24-30, 36, 37 H OMEWORK

18. L OGARITHMIC F UNCTIONS Section 8.4 (Day 2) RUL E!

19. 8.4 L OGARITHMIC F UNCTIONS Objectives: 1.Write logarithmic functions in exponential form and back. 2.Evaluate logs with and without a calculator. 3.Evaluate logarithmic functions. 4.Understand logs and inverses. 5.Graph logarithmic functions. Vocabulary: logarithm, common logarithm, natural logarithm

20. From the definition of a logarithm, we noticed that the logarithmic function, g(x) = log b x, is the inverse of the exponential function f(x) = b x. Recall: How do we verify if two functions are inverses?

21. W HAT D OES T HIS M EAN ? This means that they offset each other, or they “undo” each other. These two functions are inverses to each other.

22. U SING I NVERSES : S IMPLIFY THE E XPRESSION x x x x

23. x 2x 3x

24. H OW D O W E FIND I NVERSES ? 1. Switch x and y. 2. Solve for y. 3. KAPOOYA! DONE! 4. Check using composition because we are diligent students. In General…

25. L ET ’ S L OOK A T THE S PECIFICS … In General… 1. Switch x and y. 2. Solve for y. 3. KAPOOYA! DONE! 4. Check using composition because we are diligent students.

26. L ET ’ S L OOK A T THE S PECIFICS … In General… 1. Switch x and y. 2. Solve for y. 3. KAPOOYA! DONE! 4. Check using composition because we are diligent students.

27. Examples: Find the inverse of the function a)y = log 8 x b)y = ln(x – 3) Answers: a)y = 8 x b)y = e x + 3

28. Practice: Find the inverse of a)y = log 2/5 x b)y = ln(2x – 10) Answers: a)y = (2/5) x b)y = (e x + 10)/2

29. Function Family The graph of the function y = f(x – h)  kx – h = 0, x = h is the graph of the function y = f(x) shift h unit to the right and k unit up/down. The graph of the function y = f(x + h)  kx + h = 0, x = –h is the graph of the function y = f(x) shift h unit to the left and k unit up/down.

30. Logarithmic Function Family The graph of the logarithmic function has the following characterisitcs: y = log b (x − h) + k 1.) The line x = h is a vertical asymptote. 2.) The domain is x > h, and the range is all real numbers. 3.) If b > 1, the graph moves up to the right. If 0 < b < 1, the graph moves down to the right.

31. Example: Graph the function, state domain and range. a)y = log 1/2 (x + 4) + 2 b) y = log 3 (x – 2) – 1 1- 4 12 0 0 D: x > -4, R: all real numbersD: x > 2, R: all real numbers

32. N OTICE Vertical shift Horizontal shift

33. pg. 491 #49-52, 58-63, 65-67 H OMEWORK