Which of the following is equivalent to (log_(c)(6))/(log(6)) ? Choose 1 answer: (A) log(c) (B) log_(c)(1) (C) (1)/(log(c)) (D) (1)/(log(6))

Question

Which of the following is equivalent to  (log_(c)(6))/(log(6)) ? Choose 1 answer: (A)  log(c) (B)  log_(c)(1) (C)  (1)/(log(c)) (D)  (1)/(log(6))

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Apply Change of Base Formula: We can use the change of base formula for logarithms, which states that log_(a)(b) = log(c)(b) / log(c)(a) for any positive numbers a, b, and c (where a ≠ 1 and c ≠ 1). We can apply this formula to the given expression by choosing a new base for the logarithm.Rewrite Expression: Let's rewrite the given expression using the change of base formula. We have: (log_(c)(6))/(log(6)) = log(c)(6) / log(c)(c) Since log(c)(c) = 1 (because any number to the power of 0 is 1), the expression simplifies to: log(c)(6) / 1 = log(c)(6)Simplify Expression: Now, we need to match our simplified expression to one of the given answer choices. The expression log(c)(6) does not match any of the choices directly, but we can look for a property that might help us further simplify or transform the expression.Eliminate Number 6: We notice that none of the answer choices contain the number 6, so we must find a way to eliminate it from our expression. We can use the fact that log(c)(1) = 0 for any base c (since any number to the power of 0 is 1). However, this does not help us directly with our current expression log(c)(6).Revisit Change of Base Formula: We can now see that the expression log(c)(6) cannot be simplified further to match any of the given answer choices. However, we can revisit the change of base formula and apply it in a different way: (log_(c)(6))/(log(6)) = 1 / (log(6) / log(c)(6)) = 1 / log_(6)(c) This matches one of the answer choices.Match Answer Choice: The correct answer choice that matches our final expression 1 / log_(6)(c) is: (C) (1)/(log(c)) This is because the change of base formula allows us to write the denominator log(6) / log(c)(6) as log_(6)(c), and taking the reciprocal gives us 1 / log_(6)(c).

Which of the following is equivalent to $\newline$ $\frac{\log_{c}(6)}{\log(6)}$ ? $\newline$ Choose $1$ answer: $\newline$ (A) $\log(c)$ $\newline$ (B) $\log_{c}(1)$ $\newline$ (C) $\frac{1}{\log(c)}$ $\newline$ (D) $\frac{1}{\log(6)}$

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