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Splash Screen. Five-Minute Check (over Lesson 3-2) Then/Now Key Concept: Properties of Logarithms Example 1: Use the Properties of Logarithms Example 2: Simplify Logarithms Example 3: Expand Logarithmic Expressions Example 4: Condense Logarithmic Expressions

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  1. Splash Screen

  2. Five-Minute Check (over Lesson 3-2) Then/Now Key Concept: Properties of Logarithms Example 1: Use the Properties of Logarithms Example 2: Simplify Logarithms Example 3: Expand Logarithmic Expressions Example 4: Condense Logarithmic Expressions Key Concept: Change of Base Formula Example 5: Use the Change of Base Formula Example 6: Use the Change of Base Formula Lesson Menu

  3. Evaluate . A. B. C.1 D.2 5–Minute Check 1

  4. Evaluate log5 5. A.–1 B.0 C.1 D.5 5–Minute Check 2

  5. Evaluate 10log 2. A.1 B.2 C.5 D.10 5–Minute Check 3

  6. Evaluate ln(–3). A.about –1.1 B.about 0.48 C.about 1.1 D.undefined 5–Minute Check 4

  7. A. B. C. D. A. Sketch the graph of f(x) = log3x. 5–Minute Check 5

  8. A.D: (–∞, ∞); R: (0, ∞); x-intercept: 1; Asymptote: x-axis; Increasing (–∞, ∞) ; B.D: (–∞, ∞); R: (0, ∞); x-intercept: 1; Asymptote: x-axis; Decreasing (–∞, ∞); C.D: (0, ∞); R: (–∞, ∞); x-intercept: 1; Asymptote: y-axis; Increasing (0, ∞); D.D: (0, ∞); R: (–∞, ∞); x-intercept: 1; Asymptote: y-axis; Decreasing (–∞, ∞); B. Analyze the graph of f(x) = log3x. Describe its domain, range, intercepts, asymptotes, end behavior, and where the function is increasing or decreasing. 5–Minute Check 5

  9. Evaluate eIn x. A.x B.ln e C.e D.ex 5–Minute Check 6

  10. You evaluated logarithmic expressions with different bases. (Lesson 3–2) • Apply properties of logarithms. • Apply the Change of Base Formula. Then/Now

  11. Key Concept 1

  12. Use the Properties of Logarithms A. Express log 96 in terms of log 2 and log 3. log 96 = log (25 ● 3) 96 = 25●3 = log 25 + log 3 Product Property = 5 log 2 + log 3 Power Property Answer:5 log 2 + log 3 Example 1

  13. B. Express in terms of log 2 and log 3. = log 32 – log 9 Quotient Property Use the Properties of Logarithms = log 25 – log32 25 = 32 and 32 = 9 = 5 log 2 – 2 log 3 Power Property Answer:5 log 2 – 2 log 3 Example 1

  14. Express ln in terms of ln 3 and ln 5. A. 3 ln 5 + 3 ln 3 B. ln 53 – ln 33 C. 3 ln 5 – 3 ln 3 D. 3 ln 3 – 3 ln 5 Example 1

  15. A. Evaluate . Rewrite using rational exponents. 25 = 32 Power Property of Exponents Power Property of Logarithms logx x = 1 Answer: Simplify Logarithms Example 2

  16. Simplify Logarithms B. Evaluate 3 ln e4– 2 ln e2. 3 ln e4 – 2 ln e2 = 4(3 ln e) – 2(2 ln e) Power Property of Logarithms = 12 ln e – 4 ln e Multiply. = 12(1) – 4(1) or 8 ln e = 1 Answer:8 Example 2

  17. Evaluate . A. 4 B. C. D. Example 2

  18. Expand Logarithmic Expressions A. Expand ln 4m3n5. The expression is the logarithm of the product of 4, m3, and n5. ln 4m3n5= ln 4 + ln m3 + ln n5Product Property = ln 4 + 3 ln m + 5 ln n Power Property Answer:ln 4 + 3 ln m + 5 ln n Example 3

  19. B. Expand . The expression is the logarithm of the quotient of 2x – 3 and Quotient Property Product Property Rewrite using rational exponents. Power Property Expand Logarithmic Expressions Example 3

  20. Answer: Expand Logarithmic Expressions Example 3

  21. Expand . A. 3 ln x – ln (x – 7) B. 3 ln x + ln (x – 7) C. ln (x – 7) – 3 ln x D. lnx3 – ln (x – 7) Example 3

  22. A. Condense . Power Property Quotient Property Answer: Condense Logarithmic Expressions Example 4

  23. Condense Logarithmic Expressions B. Condense 5 ln (x + 1) + 6 ln x. 5 ln (x + 1) + 6 ln x = ln (x + 1)5 + ln x6 Power Property = ln x6(x + 1)5 Product Property Answer:ln x6(x + 1)5 Example 4

  24. A. In x(x + 3) B. C. D. Condense – ln x2 + ln (x + 3) + ln x. Example 4

  25. Key Concept 2

  26. log6 4 = Change of Base Formula Use the Change of Base Formula A. Evaluate log6 4. ≈ 0.77 Use a calculator. Answer:0.77 Example 5

  27. B.Evaluate . = Change of Base Formula Use the Change of Base Formula ≈ –1.89 Use a calculator. Answer:–1.89 Example 5

  28. Evaluate . A. –2 B. –0.5 C. 0.5 D. 2 Example 5

  29. ECOLOGY Diversity in a certain ecological environment containing two different species is modeled by the function , where N1 and N2 are the numbers of each type of species found in the sample S = ( N1+ N2 ). Find the measure of diversity for environments that find 25 and 50 species in the samples. Use the Change of Base Formula Example 6

  30. D Original equation N1 = 25, N2 = 50, and S = 75 Change of Base Formula Use the Change of Base Formula Let N1 = 25, N2 = 50, and S = 75. Substitute for the values of N1, N2, and S and solve. Example 6

  31. Use the Change of Base Formula ≈ 0.918 Use a calculator. Answer:0.918 Example 6

  32. B. ECOLOGY Diversity in a certain ecological environment containing two different species is modeled by the function ,where N1 and N2 are the numbers of each type of species found in the sample S = ( N1+ N2 ). Find the measure of diversity for environments that find 10 and 60 species in the samples. Use the Change of Base Formula Example 6

  33. D Original equation N1 = 10, N2 = 60, and S = 70 Change of Base Formula Use the Change of Base Formula Let N1 = 10, N2 = 60, and S = 70. Substitute for the values of N1, N2, and S and solve. Example 6

  34. Use the Change of Base Formula ≈ 0.592 Use a calculator. Answer:0.592 Example 6

  35. PHOTOGRAPHY In photography, exposure is the amount of light allowed to strike the film. Exposure can be adjusted by the number of stops used to take a photograph. The change in the number of stops n needed is related to the change in exposure c by n = log2c. How many stops would a photographer use to get exposure? A. –2 stops B. 2 stops C. –0.5 D. 0.5 Example 6

  36. End of the Lesson

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