To rewrite the expression \( 15 \left( \frac{1}{3} \log (x+y) - \frac{1}{5} \log (x-y) \right) \) as a single logarithm, we will first distribute the 15 across the terms:
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Rewrite the expression: \[ 15 \left( \frac{1}{3} \log (x+y) - \frac{1}{5} \log (x-y) \right) = 15 \cdot \frac{1}{3} \log (x+y) - 15 \cdot \frac{1}{5} \log (x-y) \]
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Simplify the coefficients: \[ = 5 \log (x+y) - 3 \log (x-y) \]
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Use the power rule of logarithms which states that \( a \log b = \log (b^a) \): \[ = \log ((x+y)^5) - \log ((x-y)^3) \]
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Now, use the property of logarithms that states \( \log a - \log b = \log \left( \frac{a}{b} \right) \): \[ = \log \left( \frac{(x+y)^5}{(x-y)^3} \right) \]
Thus, the expression as a single logarithm is: \[ \log \left( \frac{(x+y)^5}{(x-y)^3} \right) \]
From the options provided, the correct response would be: \[ \log \left( \frac{(x+y)^5}{(x-y)^3} \right) \]
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