Problem

ofte the following expression in Expanded form
\[
\log _{6} \frac{(x-3)^{2}}{(x+3)^{5}}
\]

Answer

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Answer

Finally, we simplify the expression to get \(\boxed{\frac{2 \log _{6}(x-3)}{\log(6)} - \frac{5 \log _{6}(x+3)}{\log(6)}}\)

Steps

Step 1 :We are given the expression \(\log _{6} \frac{(x-3)^{2}}{(x+3)^{5}}\)

Step 2 :We can use the properties of logarithms to expand this expression. The logarithm of a quotient is the difference of the logarithms. The logarithm of a power is the product of the power and the logarithm.

Step 3 :Applying these properties, we get \(2\log _{6}(x-3) - 5\log _{6}(x+3)\)

Step 4 :Finally, we simplify the expression to get \(\boxed{\frac{2 \log _{6}(x-3)}{\log(6)} - \frac{5 \log _{6}(x+3)}{\log(6)}}\)

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