4.6 Apply Properties of Logarithms

Properties of Logarithms Let b, u, and v be positive numbers such that b≠1. Product Property: (addition) logb (uv) = logb u + logb v Quotient Property: (subtraction) logb (u/v) = logb u – logb v Power Property: (multiplication) logb (u)n = n logb u

Expanding Logarithms Use log53≈.683 and log57≈1.209 Approximate Approximate

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

Expanding Logarithms Completely Single logarithms with multiple operations can be expanded to multiple logarithms with the same base

Expanding Logarithms Completely Use the properties to expand logarithms. log27x3 - log2y = log27 + log2x3 – log2y = log27 + 3·log2x – log2y

log 5mn = log 5 + log m + log n log58x3 = log58 + 3·log5x Your turn! Expand: log 5mn = log 5 + log m + log n log58x3 = log58 + 3·log5x

Condensing Logarithms Logs with the same base can be combined (condensed) into “one” logarithm

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

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

OR Change of Base Formula log37 = log37 log 7 ≈ log 3 ln 7 ≈ ln 3 Use the change of base to evaluate: When the logarithm is not in base “10”, or base “e” log37 = (base 3) (change to base 10) log 7 ≈ log 3 (base 10) 1.771 log37 (base 3) (change to base e) ln 7 ≈ ln 3 (base e) 1.771 OR

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