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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

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

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

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

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

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

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

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

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

A. Sketch the graph of f (x) = log3 x. B. C. D. 5–Minute Check 5

A. Sketch the graph of f (x) = log3 x. B. C. D. 5–Minute Check 5

B. Analyze the graph of f (x) = log3 x B. Analyze the graph of f (x) = log3 x. Describe its domain, range, intercepts, asymptotes, end behavior, and where the function is increasing or decreasing. 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 (–∞, ∞); 5–Minute Check 5

B. Analyze the graph of f (x) = log3 x B. Analyze the graph of f (x) = log3 x. Describe its domain, range, intercepts, asymptotes, end behavior, and where the function is increasing or decreasing. 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 (–∞, ∞); 5–Minute Check 5

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

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

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

Key Concept 1

A. Express log 96 in terms of log 2 and log 3. 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: Example 1

A. Express log 96 in terms of log 2 and log 3. 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

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

B. Express in terms of log 2 and log 3. Use the Properties of Logarithms B. Express in terms of log 2 and log 3. = log 32 – log 9 Quotient Property = 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

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

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

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

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

3 ln e4 – 2 ln e2 = 4(3 ln e) – 2(2 ln e) Power Property of Logarithms 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: Example 2

3 ln e4 – 2 ln e2 = 4(3 ln e) – 2(2 ln e) Power Property of Logarithms 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

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

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

The expression is the logarithm of the product of 4, m3, and n5. 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 n5 Product Property = ln 4 + 3 ln m + 5 ln n Power Property Answer: Example 3

The expression is the logarithm of the product of 4, m3, and n5. 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 n5 Product Property = ln 4 + 3 ln m + 5 ln n Power Property Answer: ln 4 + 3 ln m + 5 ln n Example 3

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

Expand Logarithmic Expressions Answer: Example 3

Expand Logarithmic Expressions Answer: Example 3

Expand . A. 3 ln x – ln (x – 7) B. 3 ln x + ln (x – 7) C. ln (x – 7) – 3 ln x D. ln x3 – ln (x – 7) Example 3

Expand . A. 3 ln x – ln (x – 7) B. 3 ln x + ln (x – 7) C. ln (x – 7) – 3 ln x D. ln x3 – ln (x – 7) Example 3

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

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

5 ln (x + 1) + 6 ln x = ln (x + 1)5 + ln x6 Power Property 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: Example 4

5 ln (x + 1) + 6 ln x = ln (x + 1)5 + ln x6 Power Property 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

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

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

Key Concept 2

log 6 4 = Change of Base Formula Use the Change of Base Formula A. Evaluate log 6 4. log 6 4 = Change of Base Formula ≈ 0.77 Use a calculator. Answer: Example 5

log 6 4 = Change of Base Formula Use the Change of Base Formula A. Evaluate log 6 4. log 6 4 = Change of Base Formula ≈ 0.77 Use a calculator. Answer: 0.77 Example 5

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

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

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

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

Use the Change of Base Formula 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. Example 6

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. D Original equation N1 = 25, N2 = 50, and S = 75 Change of Base Formula Example 6

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

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

Use the Change of Base Formula 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. Example 6

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. D Original equation N1 = 10, N2 = 60, and S = 70 Change of Base Formula Example 6

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

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

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

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

Homework - # 1- 59 odd, 135-138 Logarithms Functions Task Cards w/scan Mid Chapter Quiz

End of the Lesson