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NAME _____________________________________________ DATE ____________________________ PERIOD _____________ 7-5 Study Guide and Intervention Properties of Logarithms Properties of Logarithms Properties of exponents can be used to develop the following properties of logarithms. Product Property of Logarithms For all positive numbers a, b, and x, where x ≠ 1, log 𝑥 ab = log 𝑥 a + log 𝑥 b. Quotient Property of Logarithms For all positive numbers a, b, and x, where x ≠ 1, 𝑎 log 𝑥 = log 𝑥 a – log 𝑥 b. Power Property of Logarithms For any real number p and positive numbers m and b, where b ≠ 1, log 𝑏 𝑚𝑝 = p log 𝑏 m. 𝑏 Example: Use 𝐥𝐨𝐠 𝟑 28 ≈ 3.0331 and 𝐥𝐨𝐠 𝟑 4 ≈ 1.2619 to approximate the value of each expression. a. 𝐥𝐨𝐠 𝟑 36 log 3 36 = log 3 (32 · 4) = log 3 32 + log 3 4 = 2 + log 3 4 ≈ 2 + 1.2619 ≈ 3.2619 b. 𝐥𝐨𝐠 𝟑 7 28 log 3 7 = log 3 ( ) 4 = log 3 28 – log 3 4 ≈ 3.0331 – 1.2619 ≈ 1.7712 c. 𝐥𝐨𝐠 𝟑 256 log 3 256 = log 3 (44 ) = 4 · log 3 4 ≈ 4(1.2619) ≈ 5.0476 Exercises Use 𝐥𝐨𝐠 𝟏𝟐 3 ≈ 0.4421 and 𝐥𝐨𝐠 𝟏𝟐 7 ≈ 0.7831 to approximate the value of each expression. 7 1. log12 21 2. log12 4. log12 36 5. log12 63 6. log12 8. log12 16,807 9. log12 441 7. log12 81 49 3. log12 49 3 27 49 Use 𝐥𝐨𝐠 𝟓 3 ≈ 0.6826 and 𝐥𝐨𝐠 𝟓 4 ≈ 0.8614 to approximate the value of each expression. 10. log 5 12 11. log 5 100 13. log 5 144 14. log 5 16 ̅ 16. log 5 1.3 17. log 5 16 Chapter 7 12. log 5 0.75 27 15. log 5 375 9 18. log 5 33 81 5 Glencoe Algebra 2 NAME _____________________________________________ DATE ____________________________ PERIOD _____________ 7-5 Study Guide and Intervention (continued) Properties of Logarithms Solve Logarithmic Equations You can use the properties of logarithms to solve equations involving logarithms. Example: Solve each equation. a. 2 𝐥𝐨𝐠 𝟑 x – 𝐥𝐨𝐠 𝟑 4 = 𝐥𝐨𝐠 𝟑 25 2 log 3 x – log 3 4 = log 3 25 2 log 3 𝑥 – log 3 4 = log 3 25 log 3 𝑥2 4 𝑥2 4 2 Original equation Power Property = log 3 25 Quotient Property = 25 Property of Equality for Logarithmic Functions 𝑥 = 100 x = ±10 Multiply each side by 4. Take the square root of each side. Since logarithms are undefined for x < 0, –10 is an extraneous solution. The only solution is 10. b. 𝐥𝐨𝐠 𝟐 x + 𝐥𝐨𝐠 𝟐 (x + 2) = 3 log 2 x + log 2 (x + 2) = 3 Original equation log 2 x(x + 2) = 3 Product Property 3 x(x + 2) = 2 Definition of logarithm 𝑥 2 + 2x = 8 Distributive Property 2 𝑥 + 2x – 8 = 0 (x + 4)(x – 2) = 0 x = 2 or x = –4 Subtract 8 from each side. Factor. Zero Product Property Since logarithms are undefined for x < 0, –4 is an extraneous solution. The only solution is 2. Exercises Solve each equation. Check your solutions. 1. log 5 4 + log 5 2x = log 5 24 2. 3 log 4 6 – log 4 8 = log 4 x 1 3. 2 log 6 25 + log 6 x = log 6 20 4. log 2 4 – log 2 (x + 3) = log 2 8 5. log 6 2x – log 6 3 = log 6 (x – 1) 6. 2 log 4 (x + 1) = log 4 (11 – x) 7. log 2 x – 3 log 2 5 = 2 log 2 10 8. 3 log 2 x – 2 log 2 5x = 2 9. log 3 (c + 3) – log 3 (4c – 1) = log 3 5 10. log 5 (x + 3) – log 5 (2x – 1) = 2 Chapter 7 34 Glencoe Algebra 2 NAME _____________________________________________ DATE ____________________________ PERIOD _____________ 7-5 Skills Practice Properties of Logarithms Use 𝐥𝐨𝐠 𝟐 3 ≈ 1.5850 and 𝐥𝐨𝐠 𝟐 5 ≈ 2.3219 to approximate the value of each expression. 1. log 2 25 2. log 2 27 3 5 3. log 2 5 4. log 2 5. log 2 15 6. log 2 45 7. log 2 75 8. log 2 0.6 1 3 9 9. log 2 3 10. log 2 5 Solve each equation. Check your solutions. 11. log10 27 = 3 log10 x 12. 3 log 7 4 = 2 log 7 b 13. log 4 5 + log 4 x = log 4 60 14. log 6 2c + log 6 8 = log 6 80 15. log 5 y – log 5 8 = log 5 1 16. log 2 q – log 2 3 = log 2 7 17. log 9 4 + 2 log 9 5 = log 9 w 18. 3 log 8 2 – log 8 4 = log 8 b 19. log10 x + log10 (3x – 5) = log10 2 20. log 4 x + log 4 (2x – 3) = log 4 2 21. log 3 d + log 3 3 = 3 22. log 10 y – log10 (2 – y) = 0 23. log 2 r + 2 log 2 5 = 0 24. log 2 (x + 4) – log 2 (x – 3) = 3 25. log 4 (n + 1) – log 4 (n – 2) = 1 26. log 5 10 + log 5 12 = 3 log 5 2 + log 5 a Chapter 7 34 Glencoe Algebra 2