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Which of the following is equivalent to log_(3)(a)*log(3) ? Choose 1 answer: (A) log(a) (B) log(3) (c) log(3a) (D) log_(3)(3a)

Which of the following is equivalent to log⁡3(a)⋅log⁡(3) \log _{3}(a) \cdot \log (3) ?\newlineChoose 11 answer:\newline(A) log⁡(a) \log (a) \newline(B) log⁡(3) \log (3) \newline(C) log⁡(3a) \log (3 a) \newline(D) log⁡3(3a) \log _{3}(3 a)

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Q. Which of the following is equivalent to log⁡3(a)⋅log⁡(3) \log _{3}(a) \cdot \log (3) ?\newlineChoose 11 answer:\newline(A) log⁡(a) \log (a) \newline(B) log⁡(3) \log (3) \newline(C) log⁡(3a) \log (3 a) \newline(D) log⁡3(3a) \log _{3}(3 a)
  1. Recognize properties of logarithms: Recognize the properties of logarithms that might be relevant.\newlineWe are given the expression log⁡3(a)⋅log⁡(3)\log_{3}(a)\cdot\log(3), which involves a multiplication of two logarithms. The change of base formula might be useful here, which states that log⁡b(a)=log⁡c(a)log⁡c(b)\log_{b}(a) = \frac{\log_{c}(a)}{\log_{c}(b)} for any positive aa, bb, and cc (b,c≠1b, c \neq 1).
  2. Apply change of base formula to log⁡(3)\log(3): Apply the change of base formula to log⁡(3)\log(3).\newlineWe can write log⁡(3)\log(3) as log⁡3(3)\log_3(3) using the change of base formula, where the base cc is 33. This gives us log⁡(3)=log⁡3(3)log⁡3(3)\log(3) = \frac{\log_3(3)}{\log_3(3)}.
  3. Simplify expression from Step 22: Simplify the expression obtained in Step 22.\newlineSince log⁡3(3)\log_3(3) is simply 11 (because any log base bb of bb is 11), the expression simplifies to log⁡(3)=1\log(3) = 1.
  4. Substitute simplified expression back: Substitute the simplified expression of log⁡(3)\log(3) back into the original expression.\newlineWe now have log⁡3(a)⋅log⁡(3)=log⁡3(a)⋅1\log_{3}(a)\cdot\log(3) = \log_{3}(a)\cdot 1.
  5. Simplify expression from Step 44: Simplify the expression obtained in Step 44.\newlineMultiplying anything by 11 leaves it unchanged, so log⁡3(a)⋅1=log⁡3(a)\log_{3}(a) \cdot 1 = \log_{3}(a).
  6. Match simplified expression to options: Match the simplified expression to the given options.\newlineThe expression log⁡3(a)\log_{3}(a) matches option (A) log⁡(a)\log(a) if we assume the base of the logarithm in option (A) is 33. However, this is not explicitly stated, and the base of the common logarithm is typically 1010. Therefore, the correct match is option (D) log⁡3(3a)\log_{3}(3a) if we consider the base of the logarithm in option (A) to be 1010.

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