Which of the following is equivalent to log_(c)(16)*log_(2)(c) ? Choose 1 answer: (A) 4 (B) 8 (c) log(8) (D) log_(c)(4)

Question

Which of the following is equivalent to  log_(c)(16)*log_(2)(c) ? Choose 1 answer: (A) 4 (B) 8 (c)  log(8) (D)  log_(c)(4)

Bytelearn · Accepted Answer

Recognize change of base: Recognize the use of the change of base formula. The expression log_(c)(16)*log_(2)(c) involves two different logarithmic bases, c and 2. We can use the change of base formula to rewrite one of the logarithms in terms of the other base. Change of Base Formula: log_(b)(a) = log_(k)(a) / log_(k)(b)Apply change of base to log_(2)(c): Apply the change of base formula to log_(2)(c). We can rewrite log_(2)(c) using base c as follows: log_(2)(c) = log_(c)(c) / log_(c)(2) Since log_(c)(c) = 1 (because any log base itself is 1), we have: log_(2)(c) = 1 / log_(c)(2)Substitute expression from Step 2: Substitute the expression from Step 2 into the original expression. Now we replace log_(2)(c) in the original expression with the equivalent expression from Step 2: log_(c)(16)*log_(2)(c) = log_(c)(16) * (1 / log_(c)(2))Simplify the expression: Simplify the expression. When we multiply log_(c)(16) by 1 / log_(c)(2), we get: log_(c)(16) / log_(c)(2)Recognize division of logarithms: Recognize that this is a division of logarithms with the same base. We can use the quotient property of logarithms to combine the two logarithms into one. Quotient Property: log_(b)(P/Q) = log_(b)(P) - log_(b)(Q) log_(c)(16) / log_(c)(2) = log_(c)(16/2)Calculate division inside logarithm: Calculate the division inside the logarithm. 16 divided by 2 is 8, so we have: log_(c)(16/2) = log_(c)(8)Match result with given options: Match the result with the given options. The expression log_(c)(8) matches option (C) from the given choices.

Which of the following is equivalent to $\log _{c}(16) \cdot \log _{2}(c)$ ? $\newline$ Choose $1$ answer: $\newline$ (A) $4$ $\newline$ (B) $8$ $\newline$ (C) $\log (8)$ $\newline$ (D) $\log _{c}(4)$

Full solution

More problems from Product property of logarithms

Related topics

Which of the following is equivalent to log⁡c(16)⋅log⁡2(c) \log _{c}(16) \cdot \log _{2}(c) logc​(16)⋅log2​(c) ?\newlineChoose 111 answer:\newline(A) 444\newline(B) 888\newline(C) log⁡(8) \log (8) log(8)\newline(D) log⁡c(4) \log _{c}(4) logc​(4)

Which of the following is equivalent to $\log _{c}(16) \cdot \log _{2}(c)$ ? $\newline$ Choose $1$ answer: $\newline$ (A) $4$ $\newline$ (B) $8$ $\newline$ (C) $\log (8)$ $\newline$ (D) $\log _{c}(4)$