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.
C.
B.
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 3
Product Property
= 5 log 2 + log 3
Power 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 3
Power Property
Answer: 5 log 2 – 2 log 3
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
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)
Answer: 8
Power Property
of Logarithms
= 12 ln e – 4 ln e
Multiply.
= 12(1) – 4(1) or 8
ln e = 1
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
= ln 4 + 3 ln m + 5 ln n
Answer: ln 4 + 3 ln m + 5 ln n
Product Property
Power Property
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
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
.
Power Property
Quotient 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
= ln x6(x + 1)5
Answer: ln x6(x + 1)5
Power
Property
Product
Property
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 =
≈ 0.77
Answer: 0.77
Change of Base Formula
Use a calculator.
Use the Change of Base Formula
B. Evaluate
.
=
Change of Base Formula
≈ –1.89
Use a calculator.
Answer: –1.89
Evaluate
A. –2
B. –0.5
C. 0.5
D. 2
.
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.
,
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
Use the Change of Base Formula
≈ 0.918
Answer: 0.918
Use 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.
,
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
Use the Change of Base Formula
≈ 0.592
Answer: 0.592
Use a
calculator.
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