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Understanding Logarithmic Properties: Product, Quotient, and Power Rules Explained

Explore the essential properties of logarithms: the Product Property, Quotient Property, and Power Property. This overview provides definitions and examples to illustrate how to apply these properties effectively. You'll learn how to expand and condense logarithmic expressions and apply rules for combining logarithms. Important conditions include that the base (b), and the numbers (m, n) must be positive, and b cannot equal 1. Prepare for your quiz with clear explanations and practical examples that solidify your understanding of these fundamental concepts!

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Understanding Logarithmic Properties: Product, Quotient, and Power Rules Explained

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  1. 7-5 Properties of Logarithms Rolling them out and Wrapping them up

  2. Definitions • 1. Product Property • 2. QuotientProperty • 3. Power Property • The above will be on the quiz!

  3. Product Property • b, m, & n must be positive numbers and b ≠ 1 • log bmn = log b m + log b n • Examples: • log 4 21 = log 4 (3 · 7) = log 4 3 + log 4 7 • log 3 27 = log 3 (3 * 9) = log 3 3 + log 3 9 = 1 + 2 = 3 • log 3 4x = log 3 4 + log 3 x

  4. Quotient Rule • b, m, & n must be positive numbers and b ≠ 1 • log b = log b m – log b n • Examples: • log 4 = log 4 3 – log 4 7 • log 3 = log 3 2 – log 3 x • Notice the numerator is listed first and the denominator is subtracted from it m n 3 7 2 x

  5. Power Property • b, m, & n must be positive numbers and b ≠ 1 • log bmn = n log b m • Examples: • log 4 49 = log 4 72 = 2 log 4 7 • log 2 512 = log 2 83 = 3 log 2 8 = 3 · 3 = 9

  6. Using properties to expand an expression • log 6 =log 6 5x3 – log 6 y Quotient Property = log 6 5 + log 6 x3 – log 6 y Product Property = log 6 5 + 3 log 6 x – log 6 y Power Property 5x3 y Using properties to condense an expression • 5 log 4 2 + 7 log 4 x – 4 log 4 y • log 4 25 + log 4 x7 – log 4 y4Power Property • log 4 25x7 – log 4 y4 Product Property • log4 = log 4Quotient Property & Simplify 32x7 y4 25x7 y4

  7. Change of Base Formula • log 3 8 = ≈ • ≈ 1.893 log 8 log 3 0.9031 0.4771

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