To expand the expression \(\log_{\frac{1}{2}}\left(\frac{3x^2}{2}\right)\) using the properties and rules of logarithms, we can apply the following logarithmic properties:
- \(\log_b\left(\frac{M}{N}\right) = \log_b(M) - \log_b(N)\) (the quotient rule)
- \(\log_b(M^k) = k \cdot \log_b(M)\) (the power rule)
Now, let's use these properties to expand the given logarithm:
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Start with the original expression: \[ \log_{\frac{1}{2}}\left(\frac{3x^2}{2}\right) \]
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Apply the quotient rule: \[ = \log_{\frac{1}{2}}(3x^2) - \log_{\frac{1}{2}}(2) \]
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Now, we can expand \(\log_{\frac{1}{2}}(3x^2)\) using the product rule (\(\log_b(MN) = \log_b(M) + \log_b(N)\)): \[ = \log_{\frac{1}{2}}(3) + \log_{\frac{1}{2}}(x^2) - \log_{\frac{1}{2}}(2) \]
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Now, apply the power rule to \(\log_{\frac{1}{2}}(x^2)\): \[ = \log_{\frac{1}{2}}(3) + 2\log_{\frac{1}{2}}(x) - \log_{\frac{1}{2}}(2) \]
Putting it all together, we have: \[ \log_{\frac{1}{2}}(3) + 2\log_{\frac{1}{2}}(x) - \log_{\frac{1}{2}}(2) \]
So the expanded logarithmic expression is:
\[ \log_{\frac{1}{2}}(3) + 2 \log_{\frac{1}{2}}(x) - \log_{\frac{1}{2}}(2) \]
Looking at the provided responses, the equivalent response is:
\[ \log_{\frac{1}{2}}(3) + 2 \log(x) - \log_{\frac{1}{2}}(2) \]
Thus, that corresponds to the first option you presented.