Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege

1. Chabot Mathematics §4.2 LogFunctions Bruce Mayer, PE Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu

2. 4.1 Review § • Any QUESTIONS About • §4.1 → Exponential Functions • Any QUESTIONS About HomeWork • §4.1 → HW-18

3. §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. Logarithm → What is it? • Concept: If b > 0 and b≠ 1, then y = logbx is equivalent to x = by • Symbolically The exponent is the logarithm. x = by y = logbx The base is the base of the logarithm.

5. Logarithm Illustrated • Consider the exponential function y = f(x) = 3x. Like all exponential functions, f is one-to-one. Can a formula for the inverse Function,x= g(y) be found? Need x = 3y y = 3x y≡the exponent to which we must raise 3 to get x. f−1(x) ≡the exponent to which we must raise 3 to get x.

6. Logarithm Illustrated • Now define a new symbol to replace the words “the exponent to which we must raise 3 to get x”: log3x, 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) = 3x, then f−1(x) = log3x. Note that f−1(9) = log39 = 2, as 2 is the exponent to which we raise 3 to get 9

7. Example  Evaluate Logarithms • Evaluate: a) log381 b) log31 c) log3(1/9) • Solution: • Think of log381 as the exponent to which we raise 3 to get 81. The exponent is 4. Thus, since 34 = 81, log381 = 4. • ask: “To what exponent do we raise 3 in order to get 1?” That exponent is 0. So, log31 = 0 • To what exponent do we raise 3 in order to get 1/9? Since 3−2 = 1/9 we have log3(1/9) = −2

8. The Meaning of logax • For x > 0 and aa positive constant other than 1, logax is the exponent to which a must be raised in order to get x. Thus, logax = m means am = x • or equivalently, logax is that unique exponent for which

9. Example  Exponential to Log • Write each exponential equation in logarithmic form. • Soln

10. Example  Log to Exponential • Write each logarithmic equation in exponential form • Soln

11. Example  Evaluate Logarithms • Find the value of each of the following logarithms • Solution

12. Example  Evaluate Logarithms • Solution (cont.)

13. Example  Use Log Definition • Solve each equation for x, y or z Do On Board • Solution

14. Example  Use Log Definition • Solution (cont.)

15. 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. 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. Example  Inverse Property • Evaluate: • SolutionRemember that log523 is the exponent to which 5 is raised to get 23. Raising 5 to that exponent, obtain

18. Basic Properties of Logarithms • For any base a > 0, with a ≠ 1, Discern from the Log Definition • Logaa = 1 • As 1 is the exponent to which a must be raised to obtain a (a1 = a) • Loga1 = 0 • As 0 is the exponent to which a must be raised to obtain 1 (a0 = 1)

19. Graph Logarithmic Function • Sketch the graph of y = log3x • Soln:MakeT-Table→

20. Graph Logarithmic Function • Plot the ordered pairs and connect the dots with a smooth curve to obtain the graph of y = log3x

21. Example  Graph by Inverse • Graph y = f(x) = 3x • Solution:Use Inverse Relationfor Logs & Exponentials • Reflect the graph of y = 3xacross the line y= x to obtain the graph of y= f−1(x) = log3x

22. Exponential Function f (x) = ax Logarithmic Function f (x) = logax Domain (–∞, ∞) Range (0, ∞) y-intercept is 1 No x-intercept Properties of Exponential and Logarithmic Functions Domain (0, ∞) Range (–∞, ∞) x-intercept is 1 No y-intercept x-axis (y = 0) is the horizontal asymptote y-axis (x = 0) is the vertical asymptote

23. Exponential Function f (x) = ax Logarithmic Function f (x) = logax Is one-to-one , that is, au = av if and only if u = v Properties of Exponential and Logarithmic Functions Is one-to-one, that is, logau= logav if and only if u = v Increasing if a > 1 Decreasing if0 < a < 1 Increasing if a > 1 Decreasing if0 < a < 1

24. Graphs of Logarithmic Fcns f (x) = logax (a > 1) f (x) = logax (0 < a < 1)

25. Common Logarithms • The logarithm with base 10 is called the common logarithm and is denoted by omitting the base: logx = log10x. So y = logx if and only if x = 10y • Applying the basic properties of logs • log(10) = 1 • log(1) = 0 • log(10x) = x • 10logx = x

26. LOG 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 = log1021 • On most calculators, the key for common logarithms is marked

27. LN 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 = loge 21. • On most calculators, the key for natural logarithms is marked

28. Natural Logarithms • The logarithm with base e is called the natural logarithm and is denoted by ln x. That is, ln x = loge x. So y = lnx if and only if x = ey • Applying the basic properties of logs • ln(e) = 1 • ln(1) = 0 • ln(ex) = x • elnx = x

29. 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 • I0≡ the faintest audible sound to the average human ear, which is 10−12 watts per square meter (1x10−12 W/m2).

30. 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. Example  Sound Intensity • Solution (cont.) in the dBA eqn substitute 75 for d and 10−12 for I0 and then solve for I

32. Example  Sound Intensity • Thus the Sound Intensity at 75 dB is 10−4.5 W/m2 = 10−9/2 W/m2 • Using a Scientific calculator and find that I = 3.162x10−5 W/m2= 31.6 µW/m2

33. Example  Sound Intensity • CheckIf the sound intensity is 10−4.5 W/m2, verify that the decibel reading is 75. 

34. 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. Typical Log-Confusion • Beware that Logs do NOT behave Algebraically. In General:

36. Change of Base Rule • Let a, b, and c be positive real numbers with a ≠ 1 and b ≠ 1. Then logbx can be converted to a different base as follows:

37. 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. Example  Evaluate Logs • Compute log513 by changing to (a) common logarithms (b) natural logarithms • Soln

39. Example  Evaluate Logs • Use the change-of-base formula to calculate log512. • Round the answer to four decimal places • Solution  • Check

40. Example  Evaluate Logs • Find log37 using the change-of-base formula • Solution Substituting into

41. Example  Use The Rules • Express as an equivalent expression using individual logarithms of x, y, & z = log4x3 – log4 yz • Solna) = 3log4x – log4 yz = 3log4x – (log4 y + log4z) = 3log4x – log4 y – log4z

42. Example  Use The Rules • Solnb)

43. 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. 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. 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 A0:

46. 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 A0

47. Example  Cesium-137 ½-Life • After dividing both sides by A0 • Taking the ln of Both Sides • Solving for the HalfLife • State: The HalfLife of Cesion-137 is approximately 29.7 years

48. 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. 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? • FamiliarizeIn the Compounding Eqn replace P with 25,000, r with 0.08, A with $100,000, and then simplify.

50. Example  Compound Interest • Solution Substitute. Divide. Approximate using a calculator. • State AnswerThe account balance will reach $100,000 in about 17.33 years.