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?