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.
\[
4 \log \left(x y^{2}\right)+2 \log \left(\frac{1}{x}\right)-5 \log (y)
\]

Step 1 ：Given the expression \(4 \log \left(x y^{2}\right)+2 \log \left(\frac{1}{x}\right)-5 \log (y)\)

Step 2 ：Apply the properties of logarithms to simplify the expression

Step 3 ：Using the property \(\log(a \cdot b) = \log(a) + \log(b)\), the expression \(4 \log \left(x y^{2}\right)\) can be rewritten as \(4 \log(x) + 8 \log(y)\)

Step 4 ：Using the property \(\log(a / b) = \log(a) - \log(b)\), the expression \(2 \log \left(\frac{1}{x}\right)\) can be rewritten as \(2 \log(1) - 2 \log(x)\)

Step 5 ：Combine the above steps, the expression becomes \(4 \log(x) + 8 \log(y) + 2 - 2 \log(x) - 5 \log(y)\)

Step 6 ：Simplify the expression by combining like terms, the expression becomes \(2 \log(x) + 3 \log(y)\)

Step 7 ：Final Answer: The final simplified expression is \(\boxed{2 \log(x) + 3 \log(y)}\)