To determine which expression is equivalent to [tex]\(\log _3 \frac{c}{9}\)[/tex], we'll use properties of logarithms, specifically the quotient rule for logarithms. Here's the step-by-step solution:
1. Starting Expression: We begin with [tex]\(\log _3 \frac{c}{9}\)[/tex].
2. Logarithm Quotient Rule: The quotient rule for logarithms states that [tex]\(\log_b \frac{a}{c} = \log_b a - \log_b c\)[/tex]. In this case, our base [tex]\(b\)[/tex] is 3, [tex]\(a\)[/tex] is [tex]\(c\)[/tex], and [tex]\(c\)[/tex] is 9.
3. Apply the Quotient Rule:
[tex]\[
\log _3 \frac{c}{9} = \log _3 c - \log _3(9)
\][/tex]
4. Identify Equivalent Expression: The expression [tex]\(\log _3 c - \log _3(9)\)[/tex] matches one of the given choices.
Therefore, the expression equivalent to [tex]\(\log _3 \frac{c}{9}\)[/tex] is:
[tex]\[
\log _3 c - \log _3(9)
\][/tex]
So, the correct choice is:
[tex]\[
\log _3 c - \log _3(9)
\][/tex]
Thus, among the provided options, the correct answer is:
[tex]\[
\text{\underline{Answer:}} \quad \log _3 c - \log _3(9)
\][/tex]
