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Which of the following is equivalent to
log_(a)(18)*log_(3)(a) ?
Choose 1 answer:
(A) 6
(B)
log(6)
(C)
log_(3)(18)
(D)
log_(18)(3)

Which of the following is equivalent to $\log _{a}(18) \cdot \log _{3}(a)$ ? $\newline$ Choose $1$ answer: $\newline$ (A) $6$ $\newline$ (B) $\log (6)$ $\newline$ (C) $\log _{3}(18)$ $\newline$ (D) $\log _{18}(3)$

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Product property of logarithms

Full solution

Q. Which of the following is equivalent to

\log _{a}(18) \cdot \log _{3}(a)

?

\newline

Choose

1

answer:

\newline

(A)

6

\newline

(B)

\log (6)

\newline

(C)

\log _{3}(18)

\newline

(D)

\log _{18}(3)

Consider Second Logarithm: Now, let's consider the second logarithm, $\log_{3}(a)$ . We can write it as: $\newline$ $\log_{3}(a) = \frac{\log(a)}{\log(3)}$
Multiply Expressions: Next, we multiply the two expressions we have obtained: $\newline$ $\frac{\log(18)}{\log(a)} \times \frac{\log(a)}{\log(3)}$ $\newline$ Notice that $\log(a)$ in the numerator and denominator will cancel out: $\newline$ $\frac{\log(18)}{\log(a)} \times \frac{\log(a)}{\log(3)} = \frac{\log(18)}{\log(3)}$
Apply Change of Base Formula: The expression we have now is $\log(18) / \log(3)$ , which is the change of base formula for $\log$ base $3$ of $18$ : $\log(18) / \log(3) = \log_{3}(18)$
Final Equivalent Expression: Therefore, the equivalent expression is: $\log_{3}(18)$ This corresponds to option $(C)$ from the given choices.

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