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8-4 Properties of Logarithms

8-4 Properties of Logarithms. Using the properties of logarithms. Objectives. Using the Properties of Logarithms. Vocabulary. For any positive numbers, M,N, and b: log b MN = log b M + log b N Product Property log b = log b M - log b N Quotient Property

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8-4 Properties of Logarithms

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  1. 8-4 Properties of Logarithms Using the properties of logarithms

  2. Objectives Using the Properties of Logarithms

  3. Vocabulary For any positive numbers, M,N, and b: logbMN = logbM + logbN Product Property logb = logbM - logbN Quotient Property logbMx = x logbM Power Property M N

  4. x2 y x2 y b. logb = 2 logbx– logby Quotient Property: logb = logbx2– logby Identifying the Properties of Logarithms 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

  5. 64 16 log4 64 – log4 16 = log4Quotient Property Simplifying Logarithms 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).

  6. a. log7 t u t u log7 = log7t– log7uQuotient Property Expanding Logarithms Expand each logarithm. b. log(4p3) log(4p3) = log 4 + log p3Product Property = log 4 + 3 log pPower Property

  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 Real-World Example 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. 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. l2 l0 l1 l0 Use a calculator. 4.0 Continued (continued) The decrease in loudness would be about 4 decibels.

  9. Homework 8-4 Pg 457 # 1, 2, 3, 4, 11, 12, 19, 20

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