Which of the following is equivalent to log_(7)(a)*log_(b)(7) ? Choose 1 answer: (A) log(7) (B) log(7a) (c) log_(a)(b) (D) log_(b)(a)

Question

Which of the following is equivalent to  log_(7)(a)*log_(b)(7) ? Choose 1 answer: (A)  log(7) (B)  log(7a) (c)  log_(a)(b) (D)  log_(b)(a)

Bytelearn · Accepted Answer

Recognize change of base formula: Recognize the use of the change of base formula. The expression log_(7)(a)*log_(b)(7) suggests the use of the change of base formula, which is log_(c)(a) = log_d(a) / log_d(c) for any positive numbers a, c, and d (where a, c ≠ 1).Apply change of base formula to log: Apply the change of base formula to log_(b)(7). We can rewrite log_(b)(7) using the change of base formula with a new base, which we'll choose as 'a'. This gives us log_(b)(7) = log_a(7) / log_a(b).Substitute expression from Step 2: Substitute the expression from Step 2 into the original expression. Now we replace log_(b)(7) in the original expression with the result from Step 2, which gives us log_(7)(a) * (log_a(7) / log_a(b)).Simplify the expression: Simplify the expression. We notice that log_(7)(a) and log_a(7) are inverses of each other, meaning their product is 1. So, the expression simplifies to 1 / log_a(b).Recognize simplified expression as a logarithm: Recognize the simplified expression as a logarithm. The expression 1 / log_a(b) is the same as log_(b)(a) by the definition of the reciprocal of a logarithm.Match simplified expression to answer choices: Match the simplified expression to the answer choices. The expression log_(b)(a) corresponds to answer choice (D).

Which of the following is equivalent to $\log _{7}(a) \cdot \log _{b}(7)$ ? $\newline$ Choose $1$ answer: $\newline$ (A) $\log (7)$ $\newline$ (B) $\log (7 a)$ $\newline$ (C) $\log _{a}(b)$ $\newline$ (D) $\log _{b}(a)$

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