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Write the expression below as a single logarithm in simplest form. 2log_(b)3-log_(b)9 Answer: log_(b)(◻)

Write the expression below as a single logarithm in simplest form.\newline2log⁡b3−log⁡b9 2 \log _{b} 3-\log _{b} 9 \newlineAnswer: log⁡b(□) \log _{b}(\square)

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Quotient property of logarithmsright chevron icon

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Q. Write the expression below as a single logarithm in simplest form.\newline2log⁡b3−log⁡b9 2 \log _{b} 3-\log _{b} 9 \newlineAnswer: log⁡b(□) \log _{b}(\square)
  1. Identify Properties: Identify the properties used to combine the logarithms.\newlineThe expression given is 2log⁡b3−log⁡b92\log_{b}3 - \log_{b}9. To combine these logarithms into a single logarithm, we will use the power property and the subtraction property of logarithms.\newlinePower Property: log⁡b(Pk)=k⋅log⁡b(P)\log_b(P^k) = k \cdot \log_b(P)\newlineSubtraction Property: log⁡b(P)−log⁡b(Q)=log⁡b(PQ)\log_b(P) - \log_b(Q) = \log_b\left(\frac{P}{Q}\right)
  2. Apply Power Property: Apply the power property to the first term.\newlineThe first term is 2log⁡b32\log_{b}3, which can be rewritten using the power property as log⁡b(32)\log_{b}(3^2).\newlineSo, 2log⁡b32\log_{b}3 becomes log⁡b(32)\log_{b}(3^2) or log⁡b(9)\log_{b}(9).
  3. Combine Logarithms: Combine the two logarithms using the subtraction property.\newlineNow we have log⁡b(9)\log_{b}(9) from the first term and −log⁡b9-\log_{b}9 from the second term. Using the subtraction property, we can combine these into a single logarithm:\newlinelog⁡b(9)−log⁡b9=log⁡b(99)\log_{b}(9) - \log_{b}9 = \log_{b}(\frac{9}{9}).
  4. Simplify Expression: Simplify the expression inside the logarithm.\newlineThe expression inside the logarithm is 99\frac{9}{9}, which simplifies to 11.\newlineSo, log⁡b(99)\log_{b}\left(\frac{9}{9}\right) becomes log⁡b(1)\log_{b}(1).
  5. Recognize Identity: Recognize the logarithmic identity.\newlineThe logarithm of 11 to any base is always 00.\newlineTherefore, log⁡b(1)=0\log_{b}(1) = 0.

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