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

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

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Q. Write the expression below as a single logarithm in simplest form.\newlinelog⁡b2+2log⁡b2 \log _{b} 2+2 \log _{b} 2 \newlineAnswer: log⁡b(□) \log _{b}(\square)
  1. Identify Property Used: Identify the property used to combine the logarithms.\newlineThe expression log⁡b(2)+2log⁡b(2)\log_b(2) + 2\log_b(2) involves a sum of logarithms and a multiplication by a constant within a logarithm.\newlineThe Power Property of logarithms states that nlog⁡b(x)=log⁡b(xn)n\log_b(x) = \log_b(x^n), and the Product Property states that log⁡b(x)+log⁡b(y)=log⁡b(xy)\log_b(x) + \log_b(y) = \log_b(xy).
  2. Apply Power Property: Apply the Power Property to the term 2log⁡b(2)2\log_b(2). Using the Power Property, we can rewrite 2log⁡b(2)2\log_b(2) as log⁡b(22)\log_b(2^2). log⁡b(22)=log⁡b(4)\log_b(2^2) = \log_b(4)
  3. Combine Using Product Property: Combine the logarithms using the Product Property.\newlineNow we have log⁡b(2)+log⁡b(4)\log_b(2) + \log_b(4), which can be combined using the Product Property.\newlinelog⁡b(2)+log⁡b(4)=log⁡b(2×4)\log_b(2) + \log_b(4) = \log_b(2\times4)
  4. Simplify Inside Logarithm: Simplify the expression inside the logarithm.\newlineMultiplying the numbers inside the logarithm gives us:\newlinelog⁡b(2×4)=log⁡b(8)\log_b(2\times4) = \log_b(8)

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