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Properties of Logarithms. The Product Rule. Let b , M , and N be positive real numbers with b 1. log b (MN) = log b M + log b N The logarithm of a product is the sum of the logarithms. For example, we can use the product rule to expand ln (4 x ): ln (4 x ) = ln 4 + ln x . æ .

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## Properties of Logarithms

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**The Product Rule**Let b, M, and N be positive real numbers with b 1. logb (MN) = logbM + logbN 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.**æ**ö M log ç ÷ = log M - log N è ø b N b b The Quotient Rule Let b, M and N be positive real numbers with b 1. The logarithm of a quotient is the difference of the logarithms.**The Power Rule**Let b, M, and N be positive real numbers with b = 1, and let p be any real number. log bM p = p log bM The logarithm of a number with an exponent is the product of the exponent and the logarithm of that number.**Solution**a. log4 2 + log4 32 = log4 (2 • 32) Use the product rule. = log4 64 Although we have a single logarithm, we can simplify since 43 = 64. = 3 Text Example Write as a single logarithm: a. log4 2 + log4 32**Problems**Write the following as single logarithms:**The Change-of-Base Property**For any logarithmic bases a and b, and any positive number M, 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.**Solution Because**Solution Because This means that This means that Example: Changing Base to Common Logs Use common logarithms to evaluate log5 140. Example: Changing Base to Natural Logs Use natural logarithms to evaluate log5 140.**Use logarithms to evaluate log37.**Solution: Example or so

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