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Which of the following is equivalent to (log(3))/(log_(n)(3)) ? Choose 1 answer: (A) log(n) (B) log_(3)(n) (C) (1)/(log(n)) (D) (1)/(log_(3)(n))

Which of the following is equivalent to log⁡(3)log⁡n(3) \frac{\log (3)}{\log _{n}(3)} ?\newlineChoose 11 answer:\newline(A) log⁡(n) \log (n) \newline(B) log⁡3(n) \log _{3}(n) \newline(C) 1log⁡(n) \frac{1}{\log (n)} \newline(D) 1log⁡3(n) \frac{1}{\log _{3}(n)}

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Full solution

Q. Which of the following is equivalent to log⁡(3)log⁡n(3) \frac{\log (3)}{\log _{n}(3)} ?\newlineChoose 11 answer:\newline(A) log⁡(n) \log (n) \newline(B) log⁡3(n) \log _{3}(n) \newline(C) 1log⁡(n) \frac{1}{\log (n)} \newline(D) 1log⁡3(n) \frac{1}{\log _{3}(n)}
  1. Recognize relationship and formula: Recognize the relationship between the given expression and the change of base formula.\newlineThe expression (log⁡(3))/(log⁡n(3))(\log(3))/(\log_{n}(3)) can be related to the change of base formula for logarithms, which states that log⁡b(a)=(log⁡(c)(a))/(log⁡(c)(b))\log_{b}(a) = (\log(c)(a))/(\log(c)(b)) for any positive aa, bb, and cc, where aa and bb are not equal to 11.
  2. Apply change of base formula: Apply the change of base formula to the given expression.\newlineUsing the change of base formula, we can rewrite (log⁡(3))/(log⁡n(3))(\log(3))/(\log_{n}(3)) as log⁡n(3)\log_{n}(3). This is because the expression is essentially asking for the base nn logarithm of 33, which is the definition of log⁡n(3)\log_{n}(3).
  3. Match with answer choices: Match the rewritten expression with the answer choices.\newlineThe expression log⁡n(3)\log_{n}(3) matches with choice (B) log⁡3(n)\log_{3}(n) if we consider the properties of logarithms. However, we need to verify this by considering the change of base formula correctly.
  4. Correct application of formula: Correct the application of the change of base formula.\newlineUpon re-evaluating the previous step, we realize that the change of base formula should be applied as follows: (log⁡(3))/(log⁡n(3))(\log(3))/(\log_{n}(3)) is equivalent to log⁡n(3)\log_{n}(3) using the base 1010 logarithm, which is actually log⁡3(n)\log_{3}(n) according to the change of base formula. Therefore, the correct expression is log⁡3(n)\log_{3}(n), which corresponds to choice (B).

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