1. BMayer@ChabotCollege.edu MTH15_Lec-19_sec_4-2_Logarithmic_Fcns.pptx 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics §4.2 Log Functions

2. BMayer@ChabotCollege.edu MTH15_Lec-19_sec_4-2_Logarithmic_Fcns.pptx 2 Bruce Mayer, PE Chabot College Mathematics Review §  Any QUESTIONS About §4.1 → Exponential Functions  Any QUESTIONS About HomeWork §4.1 → HW-18 4.1

3. BMayer@ChabotCollege.edu MTH15_Lec-19_sec_4-2_Logarithmic_Fcns.pptx 3 Bruce Mayer, PE Chabot College Mathematics §4.2 Learning Goals  Define and explore logarithmic functions and their properties  Use logarithms to solve exponential equations  Examine applications involving logarithms John Napier (1550-1617) Logarithm Pioneer

4. BMayer@ChabotCollege.edu MTH15_Lec-19_sec_4-2_Logarithmic_Fcns.pptx 4 Bruce Mayer, PE Chabot College Mathematics Logarithm → What is it?  Concept: If b > 0 and b ≠ 1, then y = log b x is equivalent to x = b y  Symbolically x = b y y = log b x The exponent is the logarithm. The base is the base of the logarithm.

5. BMayer@ChabotCollege.edu MTH15_Lec-19_sec_4-2_Logarithmic_Fcns.pptx 5 Bruce Mayer, PE Chabot College Mathematics Logarithm Illustrated  Consider the exponential function y = f(x) = 3 x. Like all exponential functions, f is one-to-one. Can a formula for the inverse Function, x = g(y) be found? f − 1 (x) ≡ the exponent to which we must raise 3 to get x. y = 3 x x = 3 y y ≡ the exponent to which we must raise 3 to get x. Need

6. BMayer@ChabotCollege.edu MTH15_Lec-19_sec_4-2_Logarithmic_Fcns.pptx 6 Bruce Mayer, PE Chabot College Mathematics Logarithm Illustrated  Now define a new symbol to replace the words “the exponent to which we must raise 3 to get x”: log 3 x, read “the logarithm, base 3, of x,” or “log, base 3, of x,” means “the exponent to which we raise 3 to get x.”  Thus if f(x) = 3 x, then f −1 (x) = log 3 x. Note that f −1 (9) = log 3 9 = 2, as 2 is the exponent to which we raise 3 to get 9

7. BMayer@ChabotCollege.edu MTH15_Lec-19_sec_4-2_Logarithmic_Fcns.pptx 7 Bruce Mayer, PE Chabot College Mathematics Example  Evaluate Logarithms  Evaluate: a) log 3 81 b) log 3 1 c) log 3 (1/9)  Solution: a)Think of log 3 81 as the exponent to which we raise 3 to get 81. The exponent is 4. Thus, since 3 4 = 81, log 3 81 = 4. b)ask: “To what exponent do we raise 3 in order to get 1?” That exponent is 0. So, log 3 1 = 0 c)To what exponent do we raise 3 in order to get 1/9? Since 3 −2 = 1/9 we have log 3 (1/9) = −2

8. BMayer@ChabotCollege.edu MTH15_Lec-19_sec_4-2_Logarithmic_Fcns.pptx 8 Bruce Mayer, PE Chabot College Mathematics The Meaning of log a x  For x > 0 and a a positive constant other than 1, log a x is the exponent to which a must be raised in order to get x. Thus, log a x = m means a m = x  or equivalently, log a x is that unique exponent for which

9. BMayer@ChabotCollege.edu MTH15_Lec-19_sec_4-2_Logarithmic_Fcns.pptx 9 Bruce Mayer, PE Chabot College Mathematics Example  Exponential to Log  Write each exponential equation in logarithmic form.  Soln

10. BMayer@ChabotCollege.edu MTH15_Lec-19_sec_4-2_Logarithmic_Fcns.pptx 10 Bruce Mayer, PE Chabot College Mathematics Example  Log to Exponential  Write each logarithmic equation in exponential form  Soln

11. BMayer@ChabotCollege.edu MTH15_Lec-19_sec_4-2_Logarithmic_Fcns.pptx 11 Bruce Mayer, PE Chabot College Mathematics Example  Evaluate Logarithms  Find the value of each of the following logarithms  Solution

12. BMayer@ChabotCollege.edu MTH15_Lec-19_sec_4-2_Logarithmic_Fcns.pptx 12 Bruce Mayer, PE Chabot College Mathematics Example  Evaluate Logarithms  Solution (cont.)

13. BMayer@ChabotCollege.edu MTH15_Lec-19_sec_4-2_Logarithmic_Fcns.pptx 13 Bruce Mayer, PE Chabot College Mathematics Example  Use Log Definition  Solve each equation for x, y or z  Solution

14. BMayer@ChabotCollege.edu MTH15_Lec-19_sec_4-2_Logarithmic_Fcns.pptx 14 Bruce Mayer, PE Chabot College Mathematics Example  Use Log Definition  Solution (cont.)

15. BMayer@ChabotCollege.edu MTH15_Lec-19_sec_4-2_Logarithmic_Fcns.pptx 15 Bruce Mayer, PE Chabot College Mathematics Inverse Property of Logarithms  Recall Def: For x > 0, a > 0, and a ≠ 1,  In other words, The logarithmic function is the inverse function of the exponential function; e.g.

16. BMayer@ChabotCollege.edu MTH15_Lec-19_sec_4-2_Logarithmic_Fcns.pptx 16 Bruce Mayer, PE Chabot College Mathematics Derive Change of Base Rule  Any number >1 can be used for b, but since most calculators have ln and log functions we usually change between base-e and base-10

17. BMayer@ChabotCollege.edu MTH15_Lec-19_sec_4-2_Logarithmic_Fcns.pptx 17 Bruce Mayer, PE Chabot College Mathematics Example  Inverse Property  Evaluate:  Solution Remember that log 5 23 is the exponent to which 5 is raised to get 23. Raising 5 to that exponent, obtain

18. BMayer@ChabotCollege.edu MTH15_Lec-19_sec_4-2_Logarithmic_Fcns.pptx 18 Bruce Mayer, PE Chabot College Mathematics Basic Properties of Logarithms  For any base a > 0, with a ≠ 1, Discern from the Log Definition 1.Log a a = 1 As 1 is the exponent to which a must be raised to obtain a (a 1 = a) 2.Log a 1 = 0 As 0 is the exponent to which a must be raised to obtain 1 (a 0 = 1)

19. BMayer@ChabotCollege.edu MTH15_Lec-19_sec_4-2_Logarithmic_Fcns.pptx 19 Bruce Mayer, PE Chabot College Mathematics Graph Logarithmic Function  Sketch the graph of y = log 3 x  Soln: Make T-Table → xy = log 3 x(x, y) 3 –3 = 1/27–3(1/27, –3) 3 –2 = 1/9–2(1/9, –2) 3 –3 = 1/3–1(1/3, –1) 3 0 = 10(1, 0) 3 1 = 31(3, 1) 3 2 = 92(9, 2)

20. BMayer@ChabotCollege.edu MTH15_Lec-19_sec_4-2_Logarithmic_Fcns.pptx 20 Bruce Mayer, PE Chabot College Mathematics Graph Logarithmic Function  Plot the ordered pairs and connect the dots with a smooth curve to obtain the graph of y = log 3 x

21. BMayer@ChabotCollege.edu MTH15_Lec-19_sec_4-2_Logarithmic_Fcns.pptx 21 Bruce Mayer, PE Chabot College Mathematics Example  Graph by Inverse  Graph y = f(x) = 3 x  Solution: Use Inverse Relation for Logs & Exponentials  Reflect the graph of y = 3 x across the line y = x to obtain the graph of y = f −1 (x) = log 3 x

22. BMayer@ChabotCollege.edu MTH15_Lec-19_sec_4-2_Logarithmic_Fcns.pptx 22 Bruce Mayer, PE Chabot College Mathematics Properties of Exponential and Logarithmic Functions Exponential Function f (x) = a x Logarithmic Function f (x) = log a x Domain (0, ∞) Range (–∞, ∞) Domain (–∞, ∞) Range (0, ∞) x-intercept is 1 No y-intercept y-intercept is 1 No x-intercept x-axis (y = 0) is the horizontal asymptote y-axis (x = 0) is the vertical asymptote

23. BMayer@ChabotCollege.edu MTH15_Lec-19_sec_4-2_Logarithmic_Fcns.pptx 23 Bruce Mayer, PE Chabot College Mathematics Properties of Exponential and Logarithmic Functions Exponential Function f (x) = a x Logarithmic Function f (x) = log a x Is one-to-one, that is, log a u = log a v if and only if u = v Is one-to-one, that is, a u = a v if and only if u = v Increasing if a > 1 Decreasing if 0 < a < 1

24. BMayer@ChabotCollege.edu MTH15_Lec-19_sec_4-2_Logarithmic_Fcns.pptx 24 Bruce Mayer, PE Chabot College Mathematics Graphs of Logarithmic Fcns f (x) = log a x (0 < a < 1)f (x) = log a x (a > 1)

25. BMayer@ChabotCollege.edu MTH15_Lec-19_sec_4-2_Logarithmic_Fcns.pptx 25 Bruce Mayer, PE Chabot College Mathematics Common Logarithms  The logarithm with base 10 is called the common logarithm and is denoted by omitting the base: logx = log 10 x. So y = logx if and only if x = 10 y  Applying the basic properties of logs 1.log(10) = 1 2.log(1) = 0 3.log(10 x ) = x 4.10 logx = x

26. BMayer@ChabotCollege.edu MTH15_Lec-19_sec_4-2_Logarithmic_Fcns.pptx 26 Bruce Mayer, PE Chabot College Mathematics Common Log Convention  By this Mathematics CONVENTION the abbreviation log, with no base written, is understood to mean logarithm base 10, or a common logarithm. Thus, log21 = log 10 21  On most calculators, the key for common logarithms is marked LOG

27. BMayer@ChabotCollege.edu MTH15_Lec-19_sec_4-2_Logarithmic_Fcns.pptx 27 Bruce Mayer, PE Chabot College Mathematics Natural Logarithms  Logarithms to the 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

28. BMayer@ChabotCollege.edu MTH15_Lec-19_sec_4-2_Logarithmic_Fcns.pptx 28 Bruce Mayer, PE Chabot College Mathematics Natural Logarithms  The logarithm with base e is called the natural logarithm and is denoted by ln x. That is, ln x = log e x. So y = lnx if and only if x = e y  Applying the basic properties of logs 1.ln(e) = 1 2.ln(1) = 0 3.ln(e x ) = x 4.e lnx = x

29. BMayer@ChabotCollege.edu MTH15_Lec-19_sec_4-2_Logarithmic_Fcns.pptx 29 Bruce Mayer, PE Chabot College Mathematics Example  Sound Intensity  This function is sometimes used to calculate sound intensity  Where d ≡ the intensity in decibels, I ≡ the intensity watts per unit of area I 0 ≡ the faintest audible sound to the average human ear, which is 10 −12 watts per square meter (1x10 −12 W/m 2 ).

30. BMayer@ChabotCollege.edu MTH15_Lec-19_sec_4-2_Logarithmic_Fcns.pptx 30 Bruce Mayer, PE Chabot College Mathematics Example  Sound Intensity  Use the Sound Intensity Equation (a.k.a. the “dBA” Eqn) to find the intensity level of sounds at a decibel level of 75 dB?  Solution: We need to isolate the intensity, I, in the dBA eqn

31. BMayer@ChabotCollege.edu MTH15_Lec-19_sec_4-2_Logarithmic_Fcns.pptx 31 Bruce Mayer, PE Chabot College Mathematics Example  Sound Intensity  Solution (cont.) in the dBA eqn substitute 75 for d and 10 −12 for I 0 and then solve for I

32. BMayer@ChabotCollege.edu MTH15_Lec-19_sec_4-2_Logarithmic_Fcns.pptx 32 Bruce Mayer, PE Chabot College Mathematics Example  Sound Intensity  Thus the Sound Intensity at 75 dB is 10 −4.5 W/m 2 = 10 −9/2 W/m 2  Using a Scientific calculator and find that I = 3.162x10 −5 W/m 2 = 31.6 µW/m 2

33. BMayer@ChabotCollege.edu MTH15_Lec-19_sec_4-2_Logarithmic_Fcns.pptx 33 Bruce Mayer, PE Chabot College Mathematics Example  Sound Intensity  Check If the sound intensity is 10 −4.5 W/m 2, verify that the decibel reading is 75. 

34. BMayer@ChabotCollege.edu MTH15_Lec-19_sec_4-2_Logarithmic_Fcns.pptx 34 Bruce Mayer, PE Chabot College Mathematics Summary of Log Rules  For any positive numbers M, N, and a with a ≠ 1, and whole number p Product Rule Power Rule Quotient Rule Base-to-Power Rule

35. BMayer@ChabotCollege.edu MTH15_Lec-19_sec_4-2_Logarithmic_Fcns.pptx 35 Bruce Mayer, PE Chabot College Mathematics Typical Log-Confusion  Beware  Beware that Logs do NOT behave Algebraically. In General:

36. BMayer@ChabotCollege.edu MTH15_Lec-19_sec_4-2_Logarithmic_Fcns.pptx 36 Bruce Mayer, PE Chabot College Mathematics Change of Base Rule  Let a, b, and c be positive real numbers with a ≠ 1 and b ≠ 1. Then log b x can be converted to a different base as follows:

37. BMayer@ChabotCollege.edu MTH15_Lec-19_sec_4-2_Logarithmic_Fcns.pptx 37 Bruce Mayer, PE Chabot College Mathematics Derive Change of Base Rule  Any number >1 can be used for b, but since most calculators have ln and log functions we usually change between base-e and base-10

38. BMayer@ChabotCollege.edu MTH15_Lec-19_sec_4-2_Logarithmic_Fcns.pptx 38 Bruce Mayer, PE Chabot College Mathematics Example  Evaluate Logs  Compute log 5 13 by changing to (a) common logarithms (b) natural logarithms  Soln

39. BMayer@ChabotCollege.edu MTH15_Lec-19_sec_4-2_Logarithmic_Fcns.pptx 39 Bruce Mayer, PE Chabot College Mathematics  Use the change-of-base formula to calculate log 5 12. Round the answer to four decimal places  Solution Example  Evaluate Logs  Check 

40. BMayer@ChabotCollege.edu MTH15_Lec-19_sec_4-2_Logarithmic_Fcns.pptx 40 Bruce Mayer, PE Chabot College Mathematics  Find log 3 7 using the change-of-base formula  Solution Example  Evaluate Logs Substituting into

41. BMayer@ChabotCollege.edu MTH15_Lec-19_sec_4-2_Logarithmic_Fcns.pptx 41 Bruce Mayer, PE Chabot College Mathematics Example  Use The Rules  Express as an equivalent expression using individual logarithms of x, y, & z  Soln a) = log 4 x 3 – log 4 yz = 3log 4 x – log 4 yz = 3log 4 x – (log 4 y + log 4 z) = 3log 4 x – log 4 y – log 4 z

42. BMayer@ChabotCollege.edu MTH15_Lec-19_sec_4-2_Logarithmic_Fcns.pptx 42 Bruce Mayer, PE Chabot College Mathematics Example  Use The Rules  Soln b)

43. BMayer@ChabotCollege.edu MTH15_Lec-19_sec_4-2_Logarithmic_Fcns.pptx 43 Bruce Mayer, PE Chabot College Mathematics Caveat on Log Rules  Because the product and quotient rules replace one term with two, it is often best to use the rules within parentheses, as in the previous example

44. BMayer@ChabotCollege.edu MTH15_Lec-19_sec_4-2_Logarithmic_Fcns.pptx 44 Bruce Mayer, PE Chabot College Mathematics Example  Cesium-137 ½-Life  A sample of radioactive Cesium-137 has been Stored, unused, for cancer treatment for 2.2 years. In that time, 5% of the original sample has decayed.  What is the half-life (time required to reduce the radioactive substance to half of its starting quantity) of Cesium-137?

45. BMayer@ChabotCollege.edu MTH15_Lec-19_sec_4-2_Logarithmic_Fcns.pptx 45 Bruce Mayer, PE Chabot College Mathematics Example  Cesium-137 ½-Life  SOLUTION:  Start with the math model for exponential Decay  Recall the Given information: after 2.2 years, 95% of the sample remains  Use the Model and given data to find k  Use data in Model:  Divide both sides by A 0 :

46. BMayer@ChabotCollege.edu MTH15_Lec-19_sec_4-2_Logarithmic_Fcns.pptx 46 Bruce Mayer, PE Chabot College Mathematics Example  Cesium-137 ½-Life  Now take the ln of both Sides  Using the Base-to-Power Rule  Find by Algebra  Now set the amount, A, to ½ of A 0

47. BMayer@ChabotCollege.edu MTH15_Lec-19_sec_4-2_Logarithmic_Fcns.pptx 47 Bruce Mayer, PE Chabot College Mathematics Example  Cesium-137 ½-Life  After dividing both sides by A 0  Taking the ln of Both Sides  Solving for the HalfLife  State: The HalfLife of Cesion-137 is approximately 29.7 years

48. BMayer@ChabotCollege.edu MTH15_Lec-19_sec_4-2_Logarithmic_Fcns.pptx 48 Bruce Mayer, PE Chabot College Mathematics Example  Compound Interest  In a Bank Account that Compounds CONTINUOUSLY the relationship between the $-Principal, P, deposited, the Interest rate, r, the Compounding time-period, t, and the $-Amount, A, in the Account:

49. BMayer@ChabotCollege.edu MTH15_Lec-19_sec_4-2_Logarithmic_Fcns.pptx 49 Bruce Mayer, PE Chabot College Mathematics Example  Compound Interest  If an account pays 8% annual interest, compounded continuously, how long will it take a deposit of $25,000 to produce an account balance of $100,000?  Familiarize In the Compounding Eqn replace P with 25,000, r with 0.08, A with $100,000, and then simplify.

50. BMayer@ChabotCollege.edu MTH15_Lec-19_sec_4-2_Logarithmic_Fcns.pptx 50 Bruce Mayer, PE Chabot College Mathematics Example  Compound Interest  Solution Substitute. Divide. Approximate using a calculator.  State Answer The account balance will reach $100,000 in about 17.33 years.

51. BMayer@ChabotCollege.edu MTH15_Lec-19_sec_4-2_Logarithmic_Fcns.pptx 51 Bruce Mayer, PE Chabot College Mathematics Example  Compound Interest  Check:  Because 17.33 was not the exact time, $100,007.45 is reasonable for the Chk

52. BMayer@ChabotCollege.edu MTH15_Lec-19_sec_4-2_Logarithmic_Fcns.pptx 52 Bruce Mayer, PE Chabot College Mathematics WhiteBoard Work  Problems From §4.2 P72 → Atmospheric Pressure at Altitude –See also: B. Mayer, “Small Signal Analysis of Source Vapor Control Requirements for APCVD”, IEEE Transactions on Semiconductor Manufacturing, vol. 9, no. 3, 1996, pg 355

53. BMayer@ChabotCollege.edu MTH15_Lec-19_sec_4-2_Logarithmic_Fcns.pptx 53 Bruce Mayer, PE Chabot College Mathematics All Done for Today Napier’s MasterWork Year 1619

54. BMayer@ChabotCollege.edu MTH15_Lec-19_sec_4-2_Logarithmic_Fcns.pptx 54 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics Appendix –

55. BMayer@ChabotCollege.edu MTH15_Lec-19_sec_4-2_Logarithmic_Fcns.pptx 55 Bruce Mayer, PE Chabot College Mathematics ConCavity Sign Chart abc −−−−−−++++++−−−−−−++++++ x ConCavity Form d 2 f/dx 2 Sign Critical (Break) Points InflectionNO Inflection Inflection

56. BMayer@ChabotCollege.edu MTH15_Lec-19_sec_4-2_Logarithmic_Fcns.pptx 56 Bruce Mayer, PE Chabot College Mathematics

57. BMayer@ChabotCollege.edu MTH15_Lec-19_sec_4-2_Logarithmic_Fcns.pptx 57 Bruce Mayer, PE Chabot College Mathematics

58. BMayer@ChabotCollege.edu MTH15_Lec-19_sec_4-2_Logarithmic_Fcns.pptx 58 Bruce Mayer, PE Chabot College Mathematics