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

Write the expression below as a single logarithm in simplest form.\newline5log⁡b2−log⁡b4 5 \log _{b} 2-\log _{b} 4 \newlineAnswer: log⁡b(□) \log _{b}(\square)

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Q. Write the expression below as a single logarithm in simplest form.\newline5log⁡b2−log⁡b4 5 \log _{b} 2-\log _{b} 4 \newlineAnswer: log⁡b(□) \log _{b}(\square)
  1. Recognize Properties: Recognize the properties of logarithms that can be applied.\newlineThe expression given is a combination of two logarithms with the same base that are being subtracted. We can use the power property of logarithms to rewrite the first term and the quotient property to combine the terms into a single logarithm.
  2. Apply Power Property: Apply the power property to the first term.\newlineThe power property of logarithms states that log⁡b(an)=n⋅log⁡b(a)\log_b(a^n) = n \cdot \log_b(a). We can apply this property to the first term to rewrite 5log⁡b25\log_{b}2 as log⁡b(25)\log_{b}(2^5).\newlineCalculation: 25=322^5 = 32
  3. Combine Using Quotient Property: Combine the two logarithms using the quotient property.\newlineThe quotient property of logarithms states that log⁡b(a)−log⁡b(c)=log⁡b(ac)\log_b(a) - \log_b(c) = \log_b\left(\frac{a}{c}\right). We can apply this property to combine log⁡b(32)\log_{b}(32) and log⁡b(4)\log_{b}(4) into a single logarithm.\newlineCalculation: log⁡b(324)\log_{b}\left(\frac{32}{4}\right)
  4. Simplify Inside Logarithm: Simplify the expression inside the logarithm.\newlineWe need to divide 3232 by 44 to simplify the expression inside the logarithm.\newlineCalculation: 32/4=832 / 4 = 8
  5. Write Final Answer: Write the final answer.\newlineThe expression 5log⁡b2−log⁡b45\log_{b}2 - \log_{b}4 can be written as a single logarithm as log⁡b(8)\log_{b}(8).

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