Five-Minute Check (over Lesson 9-3)

Main Ideas and Vocabulary

Example 1: Find Common Logarithms

Example 2: Real-World Example: Solve Logarithmic Equations

Example 3: Solve Exponential Equations Using Logarithms

Example 4: Solve Exponential Inequalities Using Logarithms

Key Concept: Change of Base Formula

Example 5: Change of Base Formula

• common logarithm

• Change of Base Formula

• Solve exponential equations and inequalities using common logarithms.

• Evaluate logarithmic expressions using the Change of Base Formula.

Find Common Logarithms

A. Use a calculator to evaluate log 6 to four decimal places.

Answer: about 0.7782

Keystrokes: ENTERLOG 6 .7781512504

Find Common Logarithms

B. Use a calculator to evaluate log 0.35 to four decimal places.

Answer: about –0.4559

Keystrokes: ENTERLOG .35 –.4559319556

A. A

B. B

C. C

D. D

0% 0%0%0%

A. 0.3010

B. 0.6990

C. 5.0000

D. 100,000.0000

A. Which value is approximately equivalent to log 5?

A. A

B. B

C. C

D. D

0% 0%0%0%

A. –0.2076

B. 0.6200

C. 1.2076

D. 4.1687

B. Which value is approximately equivalent to log 0.62?

EARTHQUAKE The amount of energy E, in ergs, that an earthquake releases is related to its Richter scale magnitude M by the equation log E = 11.8 + 1.5M. The San Fernando Valley earthquake of 1994 measured 6.6 on the Richter scale. How much energy did this earthquake release?

log E = 11.8 + 1.5MWrite the

formula.

log E = 11.8 + 1.5(6.6)Replace M

with 6.6.

log E = 21.7 Simplify.

10log E = 1021.7 Write each side using 10 as a base.

Solve Logarithmic Equations

E= 1021.7

Inverse Property of Exponents and Logarithms

Answer: The amount of energy released was about 5.01 × 1021 ergs.

Solve Logarithmic Equations

E ≈ 5.01 × 1021

Use a calculator.

1. A

2. B

3. C

4. D

0%0%0%0%

A B C D

A. –7.29 ergs

B. –2.93 ergs

C. 22.9 ergs

D. 7.94 × 1022 ergs

EARTHQUAKE The amount of energy E, in ergs, that an earthquake releases is related to its Richter scale magnitude M by the equation log E = 11.8 + 1.5M. In 1999 an earthquake in Turkey measured 7.4 on the Richter scale. How much energy did this earthquake release?

Solve 5x = 62.

5x = 62 Original equation

log 5x = log 62Property of Equality for Logarithms

x log 5= log 62Power Property of Logarithms

Answer: 2.5643

Solve Exponential Equations Using Logarithms

Divide each side by log 5.

x ≈ 2.5643 Use a calculator.

Check You can check this answer by using a calculator or by using estimation. Since 52 = 25 and 53 = 125, the value of x is between 2 and 3. Thus, 2.5643 is a reasonable solution.

Solve Exponential Equations Using Logarithms

1. A

2. B

3. C

4. D

0%0%0%0%

A B C D

A. 0.3878

B. 2.5713

C. 2.5789

D. 5.6667

What is the solution to the equation 3x = 17?

Solve 27x > 35x – 3.27x

> 35x – 3

Original inequality log 27x

> log 35x – 3

Property of Inequality for Logarithmic Functions7x log 2

> (5x – 3) log 3

Power Property of Logarithms7x log 2

> 5x log 3 – 3 log 3

Distributive Property7x log 2 – 5x log 3

> – 3 log 3

Subtract 5x log 3 from each side.

Solve Exponential Inequalities Using Logarithms

x(7 log 2 – 5 log 3) > –3 log 3 Distributive Property

Solve Exponential Inequalities Using Logarithms

Divide each side by 7 log 2 – 5 log 3.

Switch > to < because 7 log 2 – 5 log 3 is negative.Use a calculator.

Simplify.

Check: Test x = 0.

27x > 35x – 3 Original inequality

Answer: The solution set is {x | x < 5.1415}.

Solve Exponential Inequalities Using Logarithms

?27(0)> 35(0) – 3 Replace x with 0.?20 > 3–3 Simplify.

Negative Exponent Property

A. A

B. B

C. C

D. D

0% 0%0%0%

A. {x | x > –1.8233}

B. {x | x < 0.9538}

C. {x | x > –0.9538}

D. {x | x < –1.8233}

What is the solution to 53x < 10x –2?

Express log3 18 in terms of common logarithms. Then approximate its value to four decimal places.

Answer: The value of log3 18 is approximately 2.6309.

Change of Base Formula

Use a calculator.

Change of Base Formula

A. A

B. B

C. C

D. D

0% 0%0%0%

What is log5 16 expressed in terms of common logarithms and approximated to four decimal places?

A.

B.

C.

D.