Question 6 - of 10 Step 1 of 1 Use the properties of logarithms to condense the following expression as much as possible, writing the answer as a single term with a coefficient of 1 All exponents should be positive. \( 4 \log \left(x^{2} y^{3}\right)+4 \log \left(\frac{1}{x}\right)-6 \log (y) \)

To condense the expression \( 4 \log \left(x^{2} y^{3}\right)+4 \log \left(\frac{1}{x}\right)-6 \log (y) \), we can use the properties of logarithms. 1. For the first term, using the power rule, we get: \( 4 \log \left(x^{2} y^{3}\right) = \log \left((x^{2} y^{3})^{4}\right) = \log \left(x^{8} y^{12}\right) \). 2. For the second term: \( 4 \log \left(\frac{1}{x}\right) = 4 \log \left(x^{-1}\right) = \log \left((x^{-1})^{4}\right) = \log \left(x^{-4}\right) \). 3. For the third term, using the power rule again: \( -6 \log (y) = \log \left(y^{-6}\right) \). Combining these results: \[ \log \left(x^{8} y^{12}\right) + \log \left(x^{-4}\right) + \log \left(y^{-6}\right) = \log \left(x^{8} y^{12} \cdot x^{-4} \cdot y^{-6}\right) = \log \left(x^{8-4} y^{12-6}\right) = \log \left(x^{4} y^{6}\right). \] Thus, the final condensed expression is: \[ \log \left(\frac{y^{6}}{x^{4}}\right). \]