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Section 8.4

Section 8.4. Properties of Logarithms. Properties of Logarithms. ALGEBRA 2 LESSON 8-4. (For help, go to Lessons 8-3 and 1-2.). Simplify each expression. 1. log 2 4 + log 2 8 2. log 3 9 – log 3 27 3. log 2 16 ÷ log 2 64. Check Skills You’ll Need. 8-4. Properties of Logarithms.

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Section 8.4

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  1. Section 8.4 Properties of Logarithms

  2. Properties of Logarithms ALGEBRA 2 LESSON 8-4 (For help, go to Lessons 8-3 and 1-2.) Simplify each expression. 1. log24 + log28 2. log39 – log327 3. log216 ÷ log264 Check Skills You’ll Need 8-4

  3. Properties of Logarithms ALGEBRA 2 LESSON 8-4 Solutions 1. log24 = x     log28 = y 2x = 4 2y = 8 x = 2 y = 3 log24 + log28 = 2 + 3 = 5 2. log39 = x     log327 = y 3x = 9 3y = 27 x = 2 y = 3 log39 – log327 = 2 – 3 = –1 3. log216 = x     log264 = y 2x = 16 2y = 64 x = 4 y = 6 log216 ÷ log264 = 4 ÷ 6 = = 4 6 2 3 8-4

  4. x2 y x2 y b. logb = 2 logbx– logby Quotient Property: logb = logbx2– logby Properties of Logarithms ALGEBRA 2 LESSON 8-4 State the property or properties used to rewrite each expression. a. log 6 = log 2 + log 3 Product Property: log 6 = log (2•3) = log 2 + log 3 Power Property: logbx2– logby = 2 logbx– logby Quick Check 8-4

  5. 64 16 log4 64 – log4 16 = log4Quotient Property Properties of Logarithms ALGEBRA 2 LESSON 8-4 Write each logarithmic expression as a single logarithm. a. log4 64 – log4 16 = log4 4 or 1 Simplify. b. 6 log5x + log5y 6 log5x + log5y = log5 x6 + log5 y Power Property = log5 (x6y) Product Property So log4 64 – log4 16 = log4 4, and 6 log5x + log2y = log5 (x6y). Quick Check 8-4

  6. a. log7 t u t u log7 = log7t– log7uQuotient Property Properties of Logarithms ALGEBRA 2 LESSON 8-4 Expand each logarithm. b. log(4p3) log(4p3) = log 4 + log p3Product Property = log 4 + 3 log pPower Property Quick Check 8-4

  7. Relate: The reduced intensity is 40% of the present intensity. Define: Let l1 = present intensity. Let l2 = reduced intensity. Let L1 = present loudness. Let L2 = reduced loudness. Write: l2 = 0.04 l1 L1 = 10 log L2 = 10 log l1 l0 l2 l0 Properties of Logarithms ALGEBRA 2 LESSON 8-4 Manufacturers of a vacuum cleaner want to reduce its sound intensity to 40% of the original intensity. By how many decibels would the loudness be reduced? 8-4

  8. Find the decrease in loudness L1–L2. Substitute l2 = 0.40l1. – 10 log = 10 log l1 l0 – 10 log 0.40 • = 10 log l1 l0 L1–L2 = 10 log Product Property 0.40l1 l0 – 10 log – 10 ( log 0.40 + log ) = 10 log – 10 log 0.40 – 10 log l1 l0 l1 l0 l1 l0 l1 l0 = 10 log l1 l0 Distributive Property = –10 log 0.40 Combine like terms. l1 l0 l2 l0 Use a calculator. 4.0 Properties of Logarithms ALGEBRA 2 LESSON 8-4 (continued) Quick Check The decrease in loudness would be about 4 decibels. 8-4

  9. a b 1 2 Properties of Logarithms ALGEBRA 2 LESSON 8-4 Write each expression as a single logarithm. State the property you used. 1. log 12 – log 3 2. 3 log115 + log117 Expand each logarithm. 3. logc4. log3x4 Use the properties of logarithms to evaluate each expression. 5. log 0.001 + log 100 6. logyy log 4; Quotient Property log11(53 • 7); Power Property and Product Property 4 log3x logca – logcb –1 1 2 8-4

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