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Write the expression below as a single logarithm in simplest form. 4log_(b)2+log_(b)5 Answer: log_(b)(◻)

Write the expression below as a single logarithm in simplest form.\newline4log⁡b2+log⁡b5 4 \log _{b} 2+\log _{b} 5 \newlineAnswer: log⁡b(□) \log _{b}(\square)

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Q. Write the expression below as a single logarithm in simplest form.\newline4log⁡b2+log⁡b5 4 \log _{b} 2+\log _{b} 5 \newlineAnswer: log⁡b(□) \log _{b}(\square)
  1. Identify Properties: Identify the properties used to combine the logarithms.\newlineThe expression contains two logarithms that are being added together. To combine them into a single logarithm, we can use the product property of logarithms.\newlineProduct Property: log⁡b(M)+log⁡b(N)=log⁡b(M×N)\log_b (M) + \log_b (N) = \log_b (M \times N)\newlineHowever, before we can apply the product property directly, we need to address the coefficient 44 in front of the first logarithm. This can be done using the power property of logarithms.\newlinePower Property: k×log⁡b(M)=log⁡b(Mk)k \times \log_b (M) = \log_b (M^k)
  2. Apply Power Property: Apply the power property to the term with the coefficient.\newlineWe have 4log⁡b24\log_{b}2, which can be rewritten using the power property as log⁡b(24)\log_{b}(2^4).\newlineCalculation: 24=162^4 = 16\newlineSo, 4log⁡b24\log_{b}2 becomes log⁡b16\log_{b}16.
  3. Combine Logarithms: Combine the two logarithms using the product property.\newlineNow we have log⁡b16+log⁡b5\log_{b}16 + \log_{b}5, which can be combined into a single logarithm using the product property.\newlineCalculation: log⁡b(16×5)\log_{b}(16 \times 5)\newlineSince 16×5=8016 \times 5 = 80, we get log⁡b80\log_{b}80.

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