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Chapter 8 Exponential and Logarithmic Functions

Chapter 8 Exponential and Logarithmic Functions

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Chapter 8 Exponential and Logarithmic Functions

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  1. Chapter 8 Exponential and Logarithmic Functions

  2. 8.1 Exponential Models

  3. Exponential Functions An exponential function is a function with the general form y = abx Graphing Exponential Functions What does a do? What does b do? 1. y = 3( ½ )x 2. y = 3( 2)x 3. y = 5( 2)x 4. y = 7( 2)x 5. y = 2( 1.25 )x 6. y = 2( 0.80 )x

  4. A and B A is the y-intercept B is direction GrowthDecay b > 10 < b < 1

  5. Y-Intercept and Growth vs. Decay Identify each y-intercept and whether it is a growth or decay. • Y= 3(1/4)x • Y= .5(3)x • Y = (.85)x

  6. Writing Exponential Functions Write an exponential model for a graph that includes the points (2,2) and (3,4). STAT  EDIT STAT  CALC  0:ExpReg

  7. Write an exponential model for a graph that includes the points • (2, 122.5) and (3, 857.5) • (0, 24) and (3, 8/9)

  8. Modeling Exponential Functions Suppose 20 rabbits are taken to an island. The rabbit population then triples every year. The function f(x) = 20 • 3x where x is the number of years, models this situation. What does “a” represent in this problems? “b”? How many rabbits would there be after 2 years?

  9. Intervals When something grows or decays at a particular interval, we must multiply x by the intervals’ reciprocal. EX: Suppose a population of 300 crickets doubles every 6 months. Find the number of crickets after 24 months.

  10. 8.2 Exponential Functions

  11. Exponential Function Where a = starting amount (y – intercept) b = change factor x = time

  12. Modeling Exponential Functions Suppose a Zombie virus has infected 20 people at our school. The number of zombies doubles every hour. Write an equation that models this. How many zombies are there after 5 hours?

  13. Modeling Exponential Functions Suppose a Zombie virus has infected 20 people at our school. The number of zombies doubles every 30 minutes. Write an equation that models this. How many zombies are there after 5 hours?

  14. A population of 2500 triples in size every 10 years. • What will the population be in 30 years?

  15. GrowthDecay . b > 10 < b < 1 (1 + r) (1 - r)

  16. Percent to Change Factor • Increase of 25% 2. Increase of 130% • Decrease of 30% 4. Decrease of 80%

  17. Growth Factor to Percent Find the percent increase or decease from the following exponential equations. • y = 3(.5)x • y = 2(2.3)x • y = 0.5(1.25)x

  18. Percent Increase and Decrease A dish has 212 bacteria in it. The population of bacteria will grow by 80% every day. How many bacteria will be present in 4 days?

  19. Percent Increase and Decrease The house down the street has termites in the porch. The exterminator estimated that there are about 800,000 termites eating at the porch. He said that the treatment he put on the wood would kill 40% of the termites every day. • How many termites will be eating at the porch in 3 days?

  20. Compound Interest P = starting amount R = rate n = period T = time

  21. Compound Interest P = R = n = T = Find the balance of a checking account that has $3,000 compounded annually at 14% for 4 years.

  22. Compound Interest P = R = n = T = Find the balance of a checking account that has $500 compounded semiannually at 8% for 5 years.

  23. 8.3 Logarithmic Functions

  24. Logarithmic Expressions Solve for x: • 2x = 4 • 2x = 10

  25. Logarithmic Expression A Logarithm solves for the missing exponent:

  26. Convert the following exponential functions to logarithmic Functions. 1. 42 = 16 2. 51 = 5 3. 70 = 1

  27. Log to Exp form Given the following Logarithmic Functions, Convert to Exponential Functions. 1. Log4 (1/16) = -2 2. Log255 = ½

  28. Evaluating Logarithms To evaluate a log we are trying to “find the exponent.” Ex: Log5 25 Ask yourself: 5x = 25

  29. You Try!

  30. A Common Logarithm is a logarithm that uses base 10. log 10 y = x ---- > log y = x Example: log1000

  31. Common Log The Calculator will do a Common Log for us! Find the Log: • Log100 • Log(1/10)

  32. When the base of the log is not 10, we can use a Change of Base Formula to find Logs with our calculator:

  33. You Try! Find the following Logarithms using change of base formula

  34. Graph the pair of equations • y = 2x and y = log 2x • y = 3x and y = log 3x What do you notice??

  35. Graphing Logarithmic Functions A logarithmic function is the inverse of an exponential function. The inverse of a function is the same as reflecting a function across the line y = x

  36. 8.4 Properties of Logarithms

  37. Properties of Logs

  38. Identify the Property • Log 2 8 – log 2 4 = log 2 2 • Log b x3y = 3(log b x) + log b y

  39. Simplify Each Logarithm • Log 3 20 – log 3 4 • 3(Log 2 x) + log 2 y • 3(log 2) + log 4 – log 16

  40. Expand Each Logarithm • Log 5 (x/y) • Log 3r4 • Log 2 7b

  41. 8.5 Exponential and Logarithmic Equations

  42. Remember! Exponential and Logarithmic equations are INVERSES of one another. Because of this, we can use them to solve each type of equation!

  43. Exponential Equations An Exponential Equation is an equation with an unknown for an exponent. Ex: 4x = 34

  44. Try Some! • 5x = 27 2. 73x = 20 3. 62x = 21 4. 3x+4 = 101 5. 11x-5 + 50 = 250

  45. Logarithmic Equation To Solve Logarithmic Equation we can transform them into Exponential Equations! Ex: Log (3x + 1) = 5

  46. You Try! • Log (7 – 2x) = -1 • Log ( 5 – 2x) = 0 • Log (6x) – 3 = -4

  47. Using Properties to Solve Equations Use the properties of logs to simplify logarithms first before solving! Ex: 2 log(x) – log (3) = 2

  48. You Try! • log 6 – log 3x = -2 • log 5 – log 2x = 1