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Introduction to Logarithmic Functions

Explore the key features and properties of logarithmic functions. Practice evaluating logarithmic expressions, solving exponential and logarithmic equations, and understand the relationship between logarithms and exponentials.

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Introduction to Logarithmic Functions

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  1. Warm Up

  2. Logarithmic Functions Objective DOL • SWBAT identify key features and apply properties of logarithmic functions. • Given 2 MC and 2 CR problems, students will identify key features and apply properties of logarithmic functions with 80% accuracy.

  3. How are logarithms and exponentials related? Essential Question

  4. Logarithmic to Exponential… 0 = log81 -4 = log2(1/16) • y = logbx • 0 = log81 • 1 = 80 • y = logbx • -4 = log2(1/16) • 1/16 = 2-4

  5. Quick Practice

  6. Quick Practice

  7. Exponential to Logarithmic… 103 = 1000 91/2 = 3 • y = logbx • 3 = log101000 • y = logbx • 1/2 = log93

  8. Evaluating log and ln on the Calculator • Use ln (natural log (base e)) • Use log button (common log (base 10)) • There isn’t a base 5 button, so… ?

  9. Change of Base Formula This will allow us to evaluate a logarithm with any base!

  10. Change of Base Formula • Practice

  11. Evaluate.

  12. Properties of Logarithms Think-Pair-Share: Why do these look familiar? How can we remember them?

  13. Properties of Logs/Exponents • Think/Pair/Share – What do these properties have in common with Properties of Exponents? Explain your thinking. .

  14. Properties of Logs/Exponents • Think/Pair/Share – What do these properties have in common with Properties of Exponents? Explain your thinking. .

  15. Properties of Logs/Exponents • Think/Pair/Share – What do these properties have in common with Properties of Exponents? Explain your thinking.

  16. Change of Base/Expand/Condense • Practice rewriting several logarithmic expressions using the properties (both expanding and collapsing):

  17. Which properties can you use to simplify each?

  18. Rewrite-Expand-Condense Practice

  19. Given log 3 =0.4771, log 4 = 0.6021, and log 5 = 0.6990 • Use the properties of logarithms to evaluate each expression. Show your work for each step. Example: log 12 log 12 = log 3(4) = log 3 + log 4 = 0.4772 + 0.6021 = 1.0793

  20. Given log 3 =0.4771, log 4 = 0.6021, and log 5 = 0.6990 • Use the properties of logarithms to evaluate each expression. Show your work for each step. • log 16 • log 3/5 • log 75 • log 60

  21. Solving Exponential and Logarithmic Equations • Practice

  22. Application a) What property will be used to solve this equation? Will you expand or condense? Power Property

  23. Application a) What property will be used to solve this equation? Will you expand or condense? Power Property

  24. Application

  25. Application a) What property will be used to solve this equation? Will you expand or condense? Power Property

  26. Application a) What property will be used to solve this equation? Will you expand or condense? Power Property

  27. Application Explain what happens in each step: Substitute in 300 Subtract 5 from both sides Convert to log form Change of base formula Solution

  28. Application a) What property will be used to solve this equation? Will you expand or condense? Power Property

  29. What is a logarithm? • a number for a given base is the exponent to which the base must be raised in order to produce the number

  30. Complete the table and graph the Exponential Function

  31. What are the key features? • Domain: • Range: • Y-intercept: • X-intercept: • Asymptote: • End behavior: All real numbers All positive numbers; y > 0 (0, 1) No x-intercept y = 0

  32. Now Graph the Inverse

  33. What are the key features? • Domain: • Range: • Y-intercept: • X-intercept: • Asymptote: x > 0 All real number No y-intercept (1, 0) x = 0

  34. Back to the inverse

  35. How are logarithms and exponentials related? Essential Question

  36. DOL #1

  37. DOL #2

  38. DOL #3 • Apply properties of logs to expand this logarithm and explain your reasoning.

  39. DOL #4 Maryville was founded in 1950. At that time, 500 people lived in the town. The population growth in Maryville follows the equation , where t is the number of years since 1950. a)Determine when the population had doubled since the founding. b) In what year was the population predicted to reach 25,000 people? c) What social implications could the population growth in that number of years have on the town?

  40. DOL Maryville was founded in 1950. At that time, 500 people lived in the town. The population growth in Maryville follows the equation , where t is the number of years since 1950. a)Determine when the population had doubled since the founding. t = 15.327 years so 1965 b) In what year was the population predicted to reach 25,000 people? t = 24.926 so 1974.9 Right before 1975 c) What social implications could the population growth in that number of years have on the town? Jobs, housing, schools, traffic, etc.

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