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Properties of Logarithms The Product Rule • Let b, M, and N be positive real numbers with b ≠ 1. • logb (MN) = logb M + logb N • The logarithm of a product is the sum of the logarithms. • For example, we can use the product rule to expand ln (4x): ln (4x) = ln 4 + ln x. The Quotient Rule • Let b, M and N be positive real numbers with b ≠ 1. M log b = log b M − lobb N N • The logarithm of a quotient is the difference of the logarithms. 1 The Power Rule • Let b, M, and N be positive real numbers with b = 1, and let p be any real number. • log b M p = p log b M • The logarithm of a number with an exponent is the product of the exponent and the logarithm of that number. Text Example Write as a single logarithm: a. log4 2 + log4 32 Solution a. log4 2 + log4 32 = log4 (2 • 32) = log4 64 =3 Use the product rule. Although we have a single logarithm, we can simplify since 43 = 64. Properties for Expanding Logarithmic Expressions • For M > 0 and N > 0: 1. logb (MN) = log b M + log b N M 2. logb = logb M − log b N N 3. log b M p = plog b M 2 Example • Use logarithmic properties to expand the expression as much as possible. 5x 2 log 2 = log 2 5 x 2 − log 2 3 3 Example cont. 5x 2 = log 2 5 x 2 − log 2 3 log 2 3 = log 2 5 + log 2 x 2 − log 2 3 Example cont. 5x 2 = log 2 5 x 2 − log 2 3 log 2 3 = log 2 5 + log 2 x 2 − log 2 3 = log 2 5 + 2 log 2 x − log 2 3 3 Properties for Condensing Logarithmic Expressions • For M > 0 and N > 0: 1. logb M + log b N = log b (MN) M 2. logb M − log b N = log b N 3. plog b M = logb M p The Change-of-Base Property • For any logarithmic bases a and b, and any positive number M, log b M = log a M log a b • The logarithm of M with base b is equal to the logarithm of M with any new base divided by the logarithm of b with that new base. Example Use logarithms to evaluate log37. Solution: log 7 log 3 7 = or so 10 log10 3 log 3 7 = ln 7 ln 3 log 3 7 = 1.77 4 Properties of Logarithms 5