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8.5 Properties of logarithms

Learn key properties of logarithms like product, quotient, and power properties to expand or condense logarithmic expressions effortlessly. Practice with examples and explore the change of base formula for more flexibility.

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8.5 Properties of logarithms

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  1. 8.5Properties of logarithms p. 493

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

  9. 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)

  10. 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

  11. Assignment

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