To determine which expression is equivalent to [tex]\(\log_3 (24^2)\)[/tex], let's go through the steps carefully.
First, we start with the original logarithmic expression:
[tex]\[ \log_3 (24^2) \][/tex]
We can use the property of logarithms that states [tex]\(\log_b (a^c) = c \cdot \log_b (a)\)[/tex]. In this case, [tex]\(a = 24\)[/tex], [tex]\(b = 3\)[/tex], and [tex]\(c = 2\)[/tex]. Applying this property, we get:
[tex]\[ \log_3 (24^2) = 2 \cdot \log_3 (24) \][/tex]
Thus, the expression [tex]\(\log_3 (24^2)\)[/tex] simplifies to:
[tex]\[ 2 \log_3 (24) \][/tex]
Therefore, the correct equivalent expression is:
[tex]\[ 2 \log_3 24 \][/tex]
Among the given options, the equivalent expression to [tex]\(\log_3 (24^2)\)[/tex] is:
[tex]\[ 2 \log _3 24 \][/tex]
So, the right answer is:
[tex]\[ \boxed{2 \log _3 24} \][/tex]
