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2. Lesson Menu Five-Minute Check (over Lesson 7–5) CCSS Then/Now New 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

3. Over Lesson 7–5 5-Minute Check 1 A.1.9864 B.2.3885 C.3.1547 D.4 Use log 3 4 ≈ 1.2619 and log 3 8 ≈ 1.8928 to approximate the value of log 3 32.

4. Over Lesson 7–5 5-Minute Check 2 A.–0.6309 B.0.1577 C.0.3155 D.0.4732 Use log 3 4 ≈ 1.2619 and log 3 8 ≈ 1.8928 to approximate the value of log 3. __ 1 2

5. Over Lesson 7–5 5-Minute Check 3 A.1 B.2 C.3 D.4 Solve log 5 6 + 3 log 5 x = log 5 48.

6. Over Lesson 7–5 5-Minute Check 4 A.10 B.8 C.6 D.4 Solve log 2 (n + 4) + log 2 n = 5.

7. Over Lesson 7–5 5-Minute Check 5 A.2 B.3 C.3.5 D.4 Solve log 6 16 – 2 log 6 4 = log 6 (x + 1) + log 6. __ 1 4

8. Over Lesson 7–5 5-Minute Check 6 Which of the following equations is false? A.log 8 m 5 = 5 log 8 m B.log a 6 – log a 3 = log a 2 C. D.log b 2x = log b 2 + log b x

9. CCSS Content Standards A.CED.1 Create equations and inequalities in one variable and use them to solve problems. Mathematical Practices 4 Model with mathematics.

10. Then/Now You simplified expressions and solved equations using properties of logarithms. Solve exponential equations and inequalities using common logarithms. Evaluate logarithmic expressions using the Change of Base Formula.

11. Vocabulary common logarithm Change of Base Formula

12. Example 1 Find Common Logarithms A. Use a calculator to evaluate log 6 to the nearest ten-thousandth. Answer: about 0.7782 Keystrokes: ENTER LOG 6.7781512504

13. Example 1 Find Common Logarithms B. Use a calculator to evaluate log 0.35 to the nearest ten-thousandth. Answer: about –0.4559 Keystrokes: ENTER LOG.35 –.4559319556

14. Example 1 A.0.3010 B.0.6990 C.5.0000 D.100,000.0000 A. Which value is approximately equivalent to log 5?

15. Example 1 A.–0.2076 B.0.6200 C.1.2076 D.4.1687 B. Which value is approximately equivalent to log 0.62?

16. Example 2 Solve Logarithmic Equations Original equation JET ENGINES The loudness L, in decibels, of a sound is where I is the intensity of the sound and m is the minimum intensity of sound detectable by the human ear. The sound of a jet engine can reach a loudness of 125 decibels. How many times the minimum intensity of audible sound is this, if m is defined to be 1?

17. Example 2 Solve Logarithmic Equations Exponential form Answer: The sound of a jet engine is approximately 3 × 10 12 or 3 trillion times the minimum intensity of sound detectable by the human ear. Use a calculator. I ≈ 3.162 × 10 12 Replace L with 125 and m with 1. Divide each side by 10 and simplify.

18. Example 2 A.1,585,000,000 times the minimum intensity B.1,629,000,000 times the minimum intensity C.1,912,000,000 times the minimum intensity D.2,788,000,000 times the minimum intensity DEMOLITION The loudness L, in decibels, of a sound is where I is the intensity of the sound and m is the minimum intensity of sound detectable by the human ear. Refer to Example 2. The sound of the demolition of an old building can reach a loudness of 92 decibels. How many times the minimum intensity of audible sound is this, if m is defined to be 1?

19. Example 3 Solve Exponential Equations Using Logarithms Solve 5 x = 62. Round to the nearest ten-thousandth. 5 x = 62Original equation log 5 x = log 62Property of Equality for Logarithms x log 5= log 62Power Property of Logarithms Answer: about 2.5643 x≈ 2.5643Use a calculator. Divide each side by log 5.

20. Example 3 Solve Exponential Equations Using Logarithms 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.

21. Example 3 A.x = 0.3878 B.x = 2.5713 C.x = 2.5789 D.x = 5.6667 What is the solution to the equation 3 x = 17?

22. Example 4 Solve Exponential Inequalities Using Logarithms Solve 3 7x > 2 5x – 3. Round to the nearest ten-thousandth. 3 7x > 2 5x – 3 Original inequality log 3 7x > log 2 5x – 3 Property of Inequality for Logarithmic Functions 7x log 3> (5x – 3) log 2Power Property of Logarithms

23. Example 4 Solve Exponential Inequalities Using Logarithms 7x log 3> 5x log 2 – 3 log 2 Distributive Property 7x log 3 – 5x log 2> – 3 log 2Subtract 5x log 2 from each side. x(7 log 3 – 5 log 2)> –3 log 2Distributive Property

24. x > –0.4922 Simplify. Example 4 Solve Exponential Inequalities Using Logarithms Use a calculator. Divide each side by 7 log 3 – 5 log 2.

25. Example 4 Solve Exponential Inequalities Using Logarithms Check: Test x = 0. 3 7x > 2 5x – 3 Original inequality Answer: The solution set is {x | x > –0.4922}. ? 3 7(0) > 2 5(0) – 3 Replace x with 0. ? 3 0 > 2 –3 Simplify. Negative Exponent Property

26. Example 4 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 ?

27. Concept

28. Example 5 Change of Base Formula Express log 5 140 in terms of common logarithms. Then round to the nearest ten-thousandth. Answer: The value of log 5 140 is approximately 3.0704. Use a calculator. Change of Base Formula

29. Example 5 What is log 5 16 expressed in terms of common logarithms? A. B. C. D.

30. End of the Lesson