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Lesson Menu 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

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

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

Over Lesson 3-2 5–Minute Check 2 Evaluate log 5 5. A.–1 B.0 C.1 D.5

Over Lesson 3-2 5–Minute Check 2 Evaluate log 5 5. A.–1 B.0 C.1 D.5

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

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

Over Lesson 3-2 5–Minute Check 4 Evaluate ln(–3). A.about –1.1 B.about 0.48 C.about 1.1 D.undefined

Over Lesson 3-2 5–Minute Check 4 Evaluate ln(–3). A.about –1.1 B.about 0.48 C.about 1.1 D.undefined

Over Lesson 3-2 5–Minute Check 5 A. Sketch the graph of f (x) = log 3 x. A. B. C. D.

Over Lesson 3-2 5–Minute Check 5 A. Sketch the graph of f (x) = log 3 x. A. B. C. D.

Over Lesson 3-2 5–Minute Check 5 B. Analyze the graph of f (x) = log 3 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 (–∞, ∞);

Over Lesson 3-2 5–Minute Check 5 B. Analyze the graph of f (x) = log 3 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 (–∞, ∞);

Over Lesson 3-2 5–Minute Check 6 Evaluate e In x. A.x B.ln e C.e D.e x

Over Lesson 3-2 5–Minute Check 6 Evaluate e In x. A.x B.ln e C.e D.e x

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

Key Concept 1

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

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

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

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

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

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

Example 2 Simplify Logarithms A. Evaluate. Rewrite using rational exponents. 2 5 = 32 Power Property of Exponents Power Property of Logarithms log x x = 1 Answer:

Example 2 Simplify Logarithms A. Evaluate. Rewrite using rational exponents. 2 5 = 32 Power Property of Exponents Power Property of Logarithms log x x = 1 Answer:

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

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

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

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

Example 3 Expand Logarithmic Expressions A. Expand ln 4m 3 n 5. The expression is the logarithm of the product of 4, m 3, and n 5. ln 4m 3 n 5 = ln 4 + ln m 3 + ln n 5 Product Property = ln ln m + 5 ln nPower Property Answer:

Example 3 Expand Logarithmic Expressions A. Expand ln 4m 3 n 5. The expression is the logarithm of the product of 4, m 3, and n 5. ln 4m 3 n 5 = ln 4 + ln m 3 + ln n 5 Product Property = ln ln m + 5 ln nPower Property Answer: ln ln m + 5 ln n

Example 3 Expand Logarithmic Expressions B. Expand. The expression is the logarithm of the quotient of 2x – 3 and Product Property Rewrite using rational exponents. Power Property Quotient 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 x 3 – 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 x 3 – ln (x – 7)

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

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

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

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

Example 4 Condense – ln x 2 + ln (x + 3) + ln x. A.In x(x + 3) B. C. D.

Example 4 Condense – ln x 2 + ln (x + 3) + ln x. A.In x(x + 3) B. C. D.

Key Concept 2

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

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

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

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

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

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

Example 6 Use the Change of Base Formula ECOLOGY Diversity in a certain ecological environment containing two different species is modeled by the function, where N 1 and N 2 are the numbers of each type of species found in the sample S = ( N 1 + N 2 ). Find the measure of diversity for environments that find 25 and 50 species in the samples.

Example 6 Let N 1 = 25, N 2 = 50, and S = 75. Substitute for the values of N 1, N 2, and S and solve. Use the Change of Base Formula DOriginal equation N 1 = 25, N 2 = 50, and S = 75 Change of Base Formula

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

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

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 N 1 and N 2 are the numbers of each type of species found in the sample S = ( N 1 + N 2 ). Find the measure of diversity for environments that find 10 and 60 species in the samples.

Example 6 Let N 1 = 10, N 2 = 60, and S = 70. Substitute for the values of N 1, N 2, and S and solve. Use the Change of Base Formula DOriginal equation N 1 = 10, N 2 = 60, and S = 70 Change of Base Formula

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

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

Example 6 A.–2 stops B.2 stops C.–0.5 D.0.5 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 = log 2 c. How many stops would a photographer use to get exposure?

Example 6 A.–2 stops B.2 stops C.–0.5 D.0.5 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 = log 2 c. How many stops would a photographer use to get exposure?

End of the Lesson