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Homework, page 317

Homework, page 317. Assuming x and y are positive, use properties of logarithms to write the expression as a sum or difference of logarithms or multiples of logarithms. 1. Homework, page 317. Assuming x and y are positive, use properties of logarithms to

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Homework, page 317

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  1. Homework, page 317 Assuming x and y are positive, use properties of logarithms to write the expression as a sum or difference of logarithms or multiples of logarithms. 1.

  2. Homework, page 317 Assuming x and y are positive, use properties of logarithms to write the expression as a sum or difference of logarithms or multiples of logarithms. 5.

  3. Homework, page 317 Assuming x and y are positive, use properties of logarithms to write the expression as a sum or difference of logarithms or multiples of logarithms. 9.

  4. Homework, page 317 Assuming x and y are positive, use properties of logarithms to write the expression as a single logarithm. 13.

  5. Homework, page 317 Assuming x and y are positive, use properties of logarithms to write the expression as a single logarithm. 17.

  6. Homework, page 317 Assuming x and y are positive, use properties of logarithms to write the expression as a single logarithm. 21.

  7. Homework, page 317 Use the change of base formula and your calculator to evaluate the logarithm. 25.

  8. Homework, page 317 Write the expression using only natural logarithms. 29.

  9. Homework, page 317 Write the expression using only common logarithms. 33.

  10. Homework, page 317 Write the expression using only common logarithms. 37. Prove the quotient rule of logarithms.

  11. Homework, page 317 Describe how to transform the graph g (x) = lnx in the graph of the given function. Sketch the graph and support with a grapher. 41. To transform g (x) into f (x), reflect about the x-axis and apply a stretch of

  12. Homework, page 317 Match the function with its graph. Identify the window dimensions, Xscl, and Yscl of the graph. 45. d. Window [-2, 8] by [-3. 3] Xscl = 1 and Yscl = 1.

  13. Homework, page 317 Graph the function and analyze it for domain, range, continuity, increasing or decreasing behavior, asymptotes, and end behavior. 49.

  14. Homework, page 317 53. The relationship between intensity I of light (in lumens) at a depth in Lake Erie is given by . What is the intensity at a depth of 40 ft?

  15. Homework, page 317 57. The logarithm of the product of two positive numbers is the sum of the logarithms of the number. Justify your answer. True. This is the definition of the Product Rule for Logarithms.

  16. Homework, page 317 61. a. b. c. d. e.

  17. Homework, page 317 65. Scientists have found that the pulse rate r of mammals to be a power function of their body weight w. a.Re-express the data in Table 3.22 in terms of their common logarithms and make a scatter plot of

  18. Homework, page 317 65. b. Compute the linear regression model c. Superimpose the regression curve on the scatter plot.

  19. Homework, page 317 65. d. Use the regression equation to predict the pulse rate of a 450-kg horse. Is the result close to 38 beats per minute? The result is close to 38 beats per minute.

  20. Homework, page 317 65. e. Why can we use either common or natural logarithms to re-express data that fit a power regression model? There is a linear relationship between common and natural logarithms, so either may be used for the re-expression of data.

  21. 3.5 Equation Solving and Modeling

  22. Quick Review

  23. Quick Review Solutions

  24. What you’ll learn about • Solving Exponential Equations • Solving Logarithmic Equations • Orders of Magnitude and Logarithmic Models • Newton’s Law of Cooling • Logarithmic Re-expression … and why The Richter scale, pH, and Newton’s Law of Cooling, are among the most important uses of logarithmic and exponential functions.

  25. Leading Questions If bu = bv, must u = v? If logbu = logbv, must u = v? If A is twice as big as B, do we say A is an order of magnitude larger than B?

  26. One-to-One Properties

  27. Example Solving an Exponential Equation Algebraically

  28. Example Solving an Exponential Equation Graphically

  29. Example Solving a Logarithmic Equation Algebraically

  30. Example Solving a Logarithmic Equation Graphically

  31. Orders of Magnitude The common logarithm of a positive quantity is its order of magnitude. Orders of magnitude can be used to compare any like quantities: • A kilometer is 3 orders of magnitude longer than a meter. • A dollar is 2 orders of magnitude greater than a penny. • New York City with 8 million people is 6 orders of magnitude bigger than Earmuff Junction with a population of 8.

  32. Richter Scale

  33. Example Comparing Magnitudes of Earthquakes Measured on the Richter Scale

  34. pH In chemistry, the acidity of a water-based solution is measured by the concentration of hydrogen ions in the solution (in moles per liter). The hydrogen-ion concentration is written [H+]. The measure of acidity used is pH, the opposite of the common log of the hydrogen-ion concentration: pH = – log [H+] More acidic solutions have higher hydrogen-ion concentrations and lower pH values.

  35. Example Using pH Measurements to Compare Hydrogen Ion concentrations Compare the hydrogen ion concentrations of vinegar, with a pH of 2.4 and salt water with a pH of 7.

  36. Newton’s Law of Cooling

  37. Example Newton’s Law of Cooling A hard-boiled egg at temperature 100ºC is placed in 15ºC water to cool. Five minutes later the temperature of the egg is 55ºC. When will the egg be 25ºC?

  38. Regression Models Related by Logarithmic Re-Expression Linear regression: y = ax + b Natural logarithmic regression: y = a + b·ln x Exponential regression: y = a·bx Power regression: y = a·xb

  39. Three Types of Logarithmic Re-Expression

  40. Three Types of Logarithmic Re-Expression (cont’d)

  41. Three Types of Logarithmic Re-Expression(cont’d)

  42. Following Questions Does the amount in a savings account grow more quickly if the interest is compounded quarterly than if it is compounded annually? Do we use future value to determine how much we must make in monthly payments to accumulate some future amount? Do we use present value to determine the monthly payment necessary to pay off a home or car loan?

  43. Homework • Review Section 3.5 • Page 331, Exercises: 1 – 61 (EOO), 73, 77

  44. 3.6 Mathematics of Finance

  45. Quick Review

  46. Quick Review Solutions

  47. What you’ll learn about • Interest Compounded Annually • Interest Compounded k Times per Year • Interest Compounded Continuously … and why The mathematics of finance is the science of letting your money work for you – valuable information indeed!

  48. Interest Compounded Annually

  49. Interest Compounded k Times per Year

  50. Example Compounding Monthly Suppose Paul invests $400 at 8% annual interest compounded monthly. Find the value of the investment after 5 years.

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