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8.5 Properties of Logarithms

8.5 Properties of Logarithms. p. 493 What are the three properties of logs? How do you expand a log? Why? How do you condense a log?. Properties of Logarithms. Let b, u, and v be positive numbers such that b ≠1. Product property: log b uv = log b u + log b v Quotient property:

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8.5 Properties of Logarithms

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  1. 8.5 Properties of Logarithms p. 493 What are the three properties of logs? How do you expand a log? Why? How do you condense a log?

  2. Properties of Logarithms • Let b, u, and v be positive numbers such that b≠1. • Product property: • logbuv = logbu + logbv • Quotient property: • logbu/v = logbu– logbv • Power property: • logbun = n logbu

  3. Use log53≈.683 and log57≈1.209 • Approximate: • log53/7 = • log53 – log57 ≈ • .683 – 1.209 = • -.526 • log521 = • log5(3·7)= • log53 + log57≈ • .683 + 1.209 = • 1.892

  4. Use log53≈.683 and log57≈1.209 • Approximate: • log549 = • log572 = • 2 log57 ≈ • 2(1.209)= • 2.418

  5. Expanding Logarithms • You can use the properties to expand logarithms. • log2 = • log27x3 - log2y = • log27 + log2x3 – log2y = • log27 + 3·log2x – log2y

  6. Your turn! • Expand: • log 5mn= • log 5 + logm + logn • Expand: • log58x3 = • log58 + 3·log5x

  7. Condensing Logarithms • log 6 + 2 log2 – log 3 = • log 6 + log 22 – log 3 = • log (6·22) – log 3 = • log = • log 8

  8. What are the three properties of logs? Product—expanded add each, Quotient—expand subtract, Power—expanded goes in front of log. • How do you expand a log? Why? Use “logb” before each addition or subtraction change. Power property will bring down exponents so you can solve for variables. • How do you condense a log? Change any addition to multiplication, subtraction to division and multiplication to power. Use one “logb”

  9. Assignment p. 496 15-55 odd Skip 29

  10. Properties of LogarithmsDay 2 • What is the change of base formula? • What is its purpose?

  11. Your turn again! • Condense: • log57 + 3·log5t = • log57t3 • Condense: • 3log2x – (log24 + log2y)= • log2

  12. Change of base formula: • u, b, and c are positive numbers with b≠1 and c≠1. Then: • logcu = • logcu = (base 10) • logcu = (base e)

  13. Examples: • Use the change of base to evaluate: • log37 = • (base 10) • log 7 ≈ • log 3 • 1.771 • (base e) • ln 7≈ • ln 3 • 1.771

  14. What is the change of base formula? What is its purpose? Lets you change on base other than 10 or e to common or natural log.

  15. Assignment Page 496, 30-56 even, 59-73 odd

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