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Lesson 4 Menu 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

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

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Lesson 4 Ex1 Find Common Logarithms A. Use a calculator to evaluate log 6 to four decimal places. Answer: about 0.7782 Keystrokes: ENTER LOG 6.7781512504

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

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A.A B.B C.C D.D Lesson 4 CYP1 A.0.3010 B.0.6990 C.5.0000 D.100,000.0000 A. Which value is approximately equivalent to log 5?

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A.A B.B C.C D.D Lesson 4 CYP1 A.–0.2076 B.0.6200 C.1.2076 D.4.1687 B. Which value is approximately equivalent to log 0.62?

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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. 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 = 21.7Simplify. 10 log E = 10 21.7 Write each side using 10 as a base. Solve Logarithmic Equations

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Lesson 4 Ex2 E= 10 21.7 Inverse Property of Exponents and Logarithms Answer: The amount of energy released was about 5.01 × 10 21 ergs. Solve Logarithmic Equations E5.01 × 10 21 Use a calculator.

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Lesson 4 CYP2 1.A 2.B 3.C 4.D A.–7.29 ergs B.–2.93 ergs C.22.9 ergs D.7.94 × 10 22 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?

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Lesson 4 Ex3 Solve 5 x = 62. 5 x = 62Original equation log 5 x = 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 2.5643Use a calculator.

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Lesson 4 Ex3 CheckYou can check this answer by using a calculator or by using estimation. Since 5 2 = 25 and 5 3 = 125, the value of x is between 2 and 3. Thus, 2.5643 is a reasonable solution. Solve Exponential Equations Using Logarithms

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1.A 2.B 3.C 4.D Lesson 4 CYP3 A.0.3878 B.2.5713 C.2.5789 D.5.6667 What is the solution to the equation 3 x = 17?

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

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Lesson 4 Ex4 x(7 log 2 – 5 log 3)> –3 log 3Distributive 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.

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Lesson 4 Ex4 Check: Test x = 0. 2 7x > 3 5x – 3 Original inequality Answer: The solution set is {x | x < 5.1415}. Solve Exponential Inequalities Using Logarithms ? 2 7(0) > 3 5(0) – 3 Replace x with 0. ? 2 0 > 3 –3 Simplify. Negative Exponent Property

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A.A B.B C.C D.D Lesson 4 CYP4 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 5 3x < 10 x –2 ?

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Lesson 4 KC1

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Lesson 4 Ex5 Express log 3 18 in terms of common logarithms. Then approximate its value to four decimal places. Answer: The value of log 3 18 is approximately 2.6309. Change of Base Formula Use a calculator. Change of Base Formula

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A.A B.B C.C D.D Lesson 4 CYP5 What is log 5 16 expressed in terms of common logarithms and approximated to four decimal places? A. B. C. D.

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End of Lesson 4

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