To determine which of the given expressions is equivalent to the logarithmic expression [tex]\(\log_3 \frac{5}{x^2}\)[/tex], we can use the properties of logarithms. We will break down the expression step by step.
1. Logarithms of Quotients:
The logarithm of a quotient is the difference of the logarithms. This can be represented as:
[tex]\[
\log_b \left(\frac{a}{c}\right) = \log_b a - \log_b c
\][/tex]
Applying this property to our expression [tex]\(\log_3 \frac{5}{x^2}\)[/tex], we get:
[tex]\[
\log_3 \frac{5}{x^2} = \log_3 5 - \log_3 x^2
\][/tex]
2. Logarithms of Powers:
The logarithm of a power is the exponent times the logarithm of the base. This can be represented as:
[tex]\[
\log_b (c^d) = d \cdot \log_b c
\][/tex]
Applying this property to [tex]\(\log_3 x^2\)[/tex], we get:
[tex]\[
\log_3 x^2 = 2 \cdot \log_3 x
\][/tex]
3. Substitution:
Now we substitute back into the expression from step 1:
[tex]\[
\log_3 \frac{5}{x^2} = \log_3 5 - \log_3 x^2 = \log_3 5 - 2 \cdot \log_3 x
\][/tex]
Therefore, the equivalent expression is:
[tex]\[
\log_3 5 - 2 \log_3 x
\][/tex]
which matches choice B.
