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 Lesson 4 Menu

Solve exponential equations and inequalities using common logarithms. Evaluate logarithmic expressions using the Change of Base Formula. common logarithm Change of Base Formula Lesson 4 MI/Vocab

Find Common Logarithms A. Use a calculator to evaluate log 6 to four decimal places. Keystrokes: ENTER LOG 6 .7781512504 Answer: about 0.7782 Lesson 4 Ex1

Find Common Logarithms B. Use a calculator to evaluate log 0.35 to four decimal places. Keystrokes: ENTER LOG .35 –.4559319556 Answer: about –0.4559 Lesson 4 Ex1

A. Which value is approximately equivalent to log 5? B. 0.6990 C. 5.0000 D. 100,000.0000 A B C D Lesson 4 CYP1

B. Which value is approximately equivalent to log 0.62? D. 4.1687 A B C D Lesson 4 CYP1

Solve Logarithmic Equations 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 + 1.5M Write 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. Lesson 4 Ex2

Solve Logarithmic Equations E = 1021.7 Inverse Property of Exponents and Logarithms E ≈ 5.01 × 1021 Use a calculator. Answer: The amount of energy released was about 5.01 × 1021 ergs. Lesson 4 Ex2

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? A. –7.29 ergs B. –2.93 ergs C. 22.9 ergs D. 7.94 × 1022 ergs A B C D Lesson 4 CYP2

Solve Exponential Equations Using Logarithms Solve 5x = 62. 5x = 62 Original equation log 5x = log 62 Property of Equality for Logarithms x log 5 = log 62 Power Property of Logarithms Divide each side by log 5. x ≈ 2.5643 Use a calculator. Answer: 2.5643 Lesson 4 Ex3

Solve Exponential Equations Using Logarithms 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.  Lesson 4 Ex3

What is the solution to the equation 3x = 17? B. 2.5713 C. 2.5789 D. 5.6667 A B C D Lesson 4 CYP3

Solve Exponential Inequalities Using Logarithms Solve 27x > 35x – 3. 27x > 35x – 3 Original inequality log 27x > log 35x – 3 Property of Inequality for Logarithmic Functions 7x log 2 > (5x – 3) log 3 Power Property of Logarithms 7x log 2 > 5x log 3 – 3 log 3 Distributive Property 7x log 2 – 5x log 3 > – 3 log 3 Subtract 5x log 3 from each side. Lesson 4 Ex4

Solve Exponential Inequalities Using Logarithms x(7 log 2 – 5 log 3) > –3 log 3 Distributive Property 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. Lesson 4 Ex4

Solve Exponential Inequalities Using Logarithms Check: Test x = 0. 27x > 35x – 3 Original inequality ? 27(0) > 35(0) – 3 Replace x with 0. ? 20 > 3–3 Simplify. Negative Exponent Property  Answer: The solution set is {x | x < 5.1415}. Lesson 4 Ex4

What is the solution to 53x < 10x –2? A. {x | x > –1.8233} B. {x | x < 0.9538} C. {x | x > –0.9538} D. {x | x < –1.8233} A B C D Lesson 4 CYP4

Lesson 4 KC1

Answer: The value of log3 18 is approximately 2.6309. Change of Base Formula Express log3 18 in terms of common logarithms. Then approximate its value to four decimal places. Change of Base Formula Use a calculator. Answer: The value of log3 18 is approximately 2.6309. Lesson 4 Ex5

What is log5 16 expressed in terms of common logarithms and approximated to four decimal places? B. C. D. A B C D Lesson 4 CYP5

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