Splash Screen. Lesson Menu Five-Minute Check (over Lesson 3-2) Then/Now Key Concept:Properties of...
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Transcript of Splash Screen. Lesson Menu Five-Minute Check (over Lesson 3-2) Then/Now Key Concept:Properties of...
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
Evaluate .
A.
B.
C. 1
D. 2
Over Lesson 3-2
Evaluate log5 5.
A. –1
B. 0
C. 1
D. 5
Over Lesson 3-2
Evaluate 10log 2.
A. 1
B. 2
C. 5
D. 10
Over Lesson 3-2
Evaluate ln(–3).
A. about –1.1
B. about 0.48
C. about 1.1
D. undefined
Over Lesson 3-2
A. Sketch the graph of f (x) = log3 x.
A.
B.
C.
D.
Over Lesson 3-2
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 (–∞, ∞);
Over Lesson 3-2
Evaluate eIn x.
A. x
B. ln e
C. e
D. ex
You evaluated logarithmic expressions with different bases. (Lesson 3–2)
• Apply properties of logarithms.
• Apply the Change of Base Formula.
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 3Product Property
= 5 log 2 + log 3Power Property
Answer: 5 log 2 + 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 3Power Property
Answer: 5 log 2 – 2 log 3
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
Express ln in terms of ln 3 and ln 5.
Simplify Logarithms
A. Evaluate .
Rewrite using rational exponents.
25 = 32
Power Property of Exponents
Power Property of Logarithms
logx x = 1
Answer:
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
Evaluate .
A. 4
B.
C.
D.
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 nPower Property
Answer: ln 4 + 3 ln m + 5 ln n
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
Expand Logarithmic Expressions
Answer:
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)
Condense Logarithmic Expressions
A. Condense .
Quotient Property
Power Property
Answer:
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
Condense – ln x2 + ln (x + 3) + ln x.
A. In x(x + 3)
B.
C.
D.
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
Use the Change of Base Formula
B. Evaluate .
= Change of Base Formula
≈ –1.89 Use a calculator.
Answer: –1.89
A. –2
B. –0.5
C. 0.5
D. 2
Evaluate .
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.
Let N1 = 25, N2 = 50, and S = 75. Substitute for the values of N1, N2, and S and solve.
Use the Change of Base Formula
D Original equation
N1 = 25, N2 = 50, and S = 75
Change of Base Formula
Answer: 0.918
Use the Change of Base Formula
≈ 0.918Use a
calculator.
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.
Let N1 = 10, N2 = 60, and S = 70. Substitute for the values of N1, N2, and S and solve.
Use the Change of Base Formula
D Original equation
N1 = 10, N2 = 60, and S = 70
Change of Base Formula
Answer: 0.592
Use the Change of Base Formula
≈ 0.592Use a
calculator.
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 = log2c. How many stops would
a photographer use to get exposure?
