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Precalculus – MAT 129

Precalculus – MAT 129. Instructor: Rachel Graham Location: BETTS Rm. 107 Time: 8 – 11:20 a.m. MWF. Chapter Three. Exponential and Logarithmic Functions. Ch. 3 Overview. Exponential Fxns and Their Graphs Logarithmic Fxns and Their Graphs Properties of Logarithms

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Precalculus – MAT 129

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  1. Precalculus – MAT 129 Instructor: Rachel Graham Location: BETTS Rm. 107 Time: 8 – 11:20 a.m. MWF

  2. Chapter Three Exponential and Logarithmic Functions

  3. Ch. 3 Overview • Exponential Fxns and Their Graphs • Logarithmic Fxns and Their Graphs • Properties of Logarithms • Solving Exponential and Logarithmic Equations • Exponential and Logarithmic Models • Nonlinear Models

  4. 3.1 – Exponential Fxns and Their Graphs • Exponential Functions • Graphs of Exponential Functions • The Natural Base e • Applications

  5. 3.1 – Exponential Functions • The exponential function f with base a is denoted by: f(x)=ax

  6. 3.1 – Graphs of Exponential Fxns • Figure 3.1 on pg. 185 shows the form of the graph of: y=ax • Figure 3.2 on pg. 185 shows the form of the graph of: y=a-x

  7. Example 1.3.1 Pg. 187 Example 4 After looking at the solution read the paragraph at the bottom of the page.

  8. 3.1 – The Natural Base e • e≈2.71828 • Useful for a base in many situations. • f(x)=ex is called the natural exponential function.

  9. Example 2.3.1 Pg. 189 Example 6 Be sure you know how to evaluate this function on your calculator.

  10. 3.1 – Applications • The most widely used application of the exponential function is for showing investment earnings with continuously compounded interest.

  11. Formulas for Compounding Interest After t years, the balance A in an account with principal P and annual interest rate r (in decimal form) is given by the following formulas: • For n compoundings per year: A=P(1+r/n)nt • For continuous compounding: A=Pert

  12. Example 3.3.1 Pg. 191 Examples 8 and 9. You will be responsible for knowing the compound interest formula.

  13. Activities (191) 1. Determine the balance A at the end of 20 years if $1500 is invested at 6.5% interest and the interest is compounded (a) quarterly and (b) continuously. 2. Determine the amount of money that should be invested at 9% interest, compounded monthly, to produce a final balance of $30,000 in 15 years.

  14. 3.2 – Logarithmic Fxns and Their Graphs • Logarithmic Functions • Graphs of Logarithmic Functions • The Natural Logarithmic Function • Applications

  15. 3.2 – Logarithmic Functions • The inverse of the exponential function is the logarithmic function. For x>0, a>0, and a≠1, y=logax if and only if x=ay. f(x)=logax is called the logarithmic function with base a.

  16. Properties of Logarithms • loga1=0 because a0=1. • logaa=1 because a1=a. • logaax =x because alogx=x. • If logax=logay, then x=y

  17. Example 1.3.2 Pg. 203 #33. Solve the equation for x. log7x=log79

  18. Solution Example 1.3.2 Pg. 203 #33. x=9

  19. 3.2 – Graphs of Logarithmic Fxns • See beige box on pg. 199

  20. 3.2 – The Natural Logarithmic Fxn For x>0, y=ln x if and only if x=ey. f(x) = logex = ln x is called the natural logarithmic function.

  21. Properties of Natural Logarithms • ln 1=0 because e0=1. • ln e=1 because e1=e. • ln ex =x because elnx=x. • If ln x=ln y, then x=y

  22. Example 2.3.2 Pg. 201 Example 9. Note both the algebraic and graphical solutions.

  23. 3.2 – Application See example 10 on pg. 202 for the best application of logarithmic functions.

  24. 3.3 – Properties of Logarithms • Change of Base • Properties of Logarithms • Rewriting Logarithmic Expressions

  25. 3.3 – Change of Base To evaluate logarithms at different bases you can use the change of base formula: logax = (logbx/ logba)

  26. Example 1.3.3 Pg. 207 Examples 1 & 2. Note both log and ln functions will yield the same result.

  27. 3.3 – Properties of Logarithms See blue box on pg. 208.

  28. Example 2.3.3 Pg. 208 Example 3 These should be pretty self explanatory.

  29. 3.3 – Rewriting Log Fxns • This is where you use the multiplication, division, and power rules to expand and condense logarithmic expressions.

  30. Example 3.3.3 Pg. 209 Examples 5&6. Note that a square root is equal to the power of ½.

  31. 3.4 – Solving Exponential and Logarithmic Equations • Introduction • Solving Exponential Equations • Solving Logarithmic Equations • Applications

  32. 3.5 –Exponential and Logarithmic Models • Introduction • Exponential Growth and Decay • Gaussian Models • Logistic Growth Models • Logarithmic Models

  33. The Models

  34. Example 1.3.5 • Example 2 on pg. 227 In a research experiment, a population of fruit flies is increasing according to the law of exponential growth. After 2 days there are 100 fruit flies, and after 4 days there are 300 fruit flies. How many flies will there be after 5 days?

  35. Example 2.3.5 • Example 5 on pg. 230 On a college campus of 5000 students, one student returns from vacation with a contagious flu virus. The spread of the virus is modeled on pg. 230 where y is the total number infected after t days. The college will cancel classes when 40% or more are infected. • How many students are infected after 5 days? • After how many days will the college cancel classes?

  36. Example 3.3.5 • On the Richter scale, the magnitude R of an earthquake of intensity I is given by R = log10 I/I0 where I0 = 1 is the minimum intensity used for comparison. Intensity is a measure of wave energy of an earthquake.

  37. Activities In Class QUIZ: pp. 234 #30, 41a, 42a.

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