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Logarithmic and Exponential Functions: Examples and Applications

This lesson covers logarithmic and exponential functions, including converting between logarithmic and exponential forms, evaluating logarithmic expressions, graphing logarithmic functions, and finding inverses of exponential functions.

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Logarithmic and Exponential Functions: Examples and Applications

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  1. Splash Screen

  2. Five-Minute Check (over Lesson 7–2) CCSS Then/Now New Vocabulary Key Concept: Logarithm with Base b Example 1: Logarithmic to Exponential Form Example 2: Exponential to Logarithmic Form Example 3: Evaluate Logarithmic Expressions Key Concept: Parent Function of Logarithmic Functions Example 4: Graph Logarithmic Functions Key Concept: Transformations of Logarithmic Functions Example 5: Graph Logarithmic Functions Example 6: Real-World Example: Find Inverses of Exponential Functions Lesson Menu

  3. A.x = –1 B.x = C.x = 1 D.x = 2 1 __ 2 Solve 42x = 163x– 1. 5-Minute Check 1

  4. A.x = –1 B.x = C.x = 1 D.x = 2 1 __ 2 Solve 42x = 163x– 1. 5-Minute Check 1

  5. Solve 8x – 1 = 2x+ 9. A.x = 10 B.x = 8 C.x = 6 D.x = 4 5-Minute Check 2

  6. Solve 8x – 1 = 2x+ 9. A.x = 10 B.x = 8 C.x = 6 D.x = 4 5-Minute Check 2

  7. Solve 52x– 7< 125. A.x < 6 B.x < 5 C.x < 4 D. x > 3 5-Minute Check 3

  8. Solve 52x– 7< 125. A.x < 6 B.x < 5 C.x < 4 D. x > 3 5-Minute Check 3

  9. Solve A.x ≥ 2 B.x ≥ 1 C.x > 0 D.x ≥ –2 5-Minute Check 4

  10. Solve A.x ≥ 2 B.x ≥ 1 C.x > 0 D.x ≥ –2 5-Minute Check 4

  11. A money market account pays 5.3% interest compounded quarterly. What will be the balance in the account after 5 years if $12,000 is invested? A. $18,360.00 B. $15,613.98 C. $15,180.00 D. $14,544.00 5-Minute Check 5

  12. A money market account pays 5.3% interest compounded quarterly. What will be the balance in the account after 5 years if $12,000 is invested? A. $18,360.00 B. $15,613.98 C. $15,180.00 D. $14,544.00 5-Minute Check 5

  13. Charlie borrowed $125,000 for his small business at a rate of 3.9% compounded annually for 30 years. At the end of the loan, how much will he have actually paid for the loan? A. $146,250 B. $271,250 C. $389,625.25 D. $393,891.35 5-Minute Check 6

  14. Charlie borrowed $125,000 for his small business at a rate of 3.9% compounded annually for 30 years. At the end of the loan, how much will he have actually paid for the loan? A. $146,250 B. $271,250 C. $389,625.25 D. $393,891.35 5-Minute Check 6

  15. Content Standards F.IF.7.e Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. F.BF.3 Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f (kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Mathematical Practices 6 Attend to precision. CCSS

  16. You found the inverse of a function. • Evaluate logarithmic expressions. • Graph logarithmic functions. Then/Now

  17. logarithm • logarithmic function Vocabulary

  18. Concept

  19. Logarithmic to Exponential Form A.Write log3 9 = 2 in exponential form. log3 9 = 2 → 9 = 32 Answer: Example 1

  20. Logarithmic to Exponential Form A.Write log3 9 = 2 in exponential form. log3 9 = 2 → 9 = 32 Answer:9 = 32 Example 1

  21. B. Write in exponential form. Logarithmic to Exponential Form Answer: Example 1

  22. B. Write in exponential form. Answer: Logarithmic to Exponential Form Example 1

  23. A. What is log2 8 = 3 written in exponential form? A. 83 = 2 B. 23 = 8 C. 32 = 8 D. 28 = 3 Example 1

  24. A. What is log2 8 = 3 written in exponential form? A. 83 = 2 B. 23 = 8 C. 32 = 8 D. 28 = 3 Example 1

  25. B. What is –2 written in exponential form? A. B. C. D. Example 1

  26. B. What is –2 written in exponential form? A. B. C. D. Example 1

  27. Exponential to Logarithmic Form A.Write 53 = 125 in logarithmic form. 53 = 125 → log5 125 = 3 Answer: Example 2

  28. Exponential to Logarithmic Form A.Write 53 = 125 in logarithmic form. 53 = 125 → log5 125 = 3 Answer:log5 125 = 3 Example 2

  29. B. Write in logarithmic form. Exponential to Logarithmic Form Answer: Example 2

  30. B. Write in logarithmic form. Answer: Exponential to Logarithmic Form Example 2

  31. A. What is 34 = 81 written in logarithmic form? A. log3 81 = 4 B. log4 81 = 3 C. log81 3 = 4 D. log3 4 = 81 Example 2

  32. A. What is 34 = 81 written in logarithmic form? A. log3 81 = 4 B. log4 81 = 3 C. log81 3 = 4 D. log3 4 = 81 Example 2

  33. B. What is written in logarithmic form? A. B. C. D. Example 2

  34. B. What is written in logarithmic form? A. B. C. D. Example 2

  35. Evaluate Logarithmic Expressions Evaluate log3 243. log3 243 = y Let the logarithm equal y. 243 = 3y Definition of logarithm 35 = 3y243 = 35 5 = y Property of Equality for Exponential Functions Answer: Example 3

  36. Evaluate Logarithmic Expressions Evaluate log3 243. log3 243 = y Let the logarithm equal y. 243 = 3y Definition of logarithm 35 = 3y243 = 35 5 = y Property of Equality for Exponential Functions Answer: So, log3 243 = 5. Example 3

  37. A. B.3 C.30 D.10,000 Evaluate log10 1000. Example 3

  38. A. B.3 C.30 D.10,000 Evaluate log10 1000. Example 3

  39. Concept

  40. Because 3 > 1, use the points (1, 0), and (b, 1). Graph Logarithmic Functions A.Graph the function f(x) = log3x. Step 1 Identify the base. b = 3 Step 2 Determine points on the graph. Step 3 Plot the points and sketch the graph. Example 4

  41. Graph Logarithmic Functions (1, 0) (b, 1) → (3, 1) Answer: Example 4

  42. Graph Logarithmic Functions (1, 0) (b, 1) → (3, 1) Answer: Example 4

  43. B.Graph the function Graph Logarithmic Functions Step 1 Identify the base. Step 2 Determine points on the graph. Example 4

  44. Graph Logarithmic Functions Step 3 Sketch the graph. Answer: Example 4

  45. Graph Logarithmic Functions Step 3 Sketch the graph. Answer: Example 4

  46. A. B. C.D. A. Graph the function f(x) = log5x. Example 4

  47. A. B. C.D. A. Graph the function f(x) = log5x. Example 4

  48. B. Graph the function . A. B. C.D. Example 4

  49. B. Graph the function . A. B. C.D. Example 4

  50. Concept

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