Use the properties of logarithms to simplify the expression by eliminating all exponents and radicals. Assume that \( x, y>0 \). \( \log \left(x^{7} y^{8}\right)= \)

To simplify the expression \( \log \left(x^{7} y^{8}\right) \), we can utilize the properties of logarithms. First, we apply the product property of logarithms, which states that the logarithm of a product is the sum of the logarithms of the individual factors. This gives us: \[ \log \left(x^{7} y^{8}\right) = \log \left(x^{7}\right) + \log \left(y^{8}\right) \] Next, we utilize the power property of logarithms, which tells us that we can bring the exponent in front of the logarithm: \[ \log \left(x^{7}\right) = 7 \log x \quad \text{and} \quad \log \left(y^{8}\right) = 8 \log y \] Putting it all together, we have: \[ \log \left(x^{7} y^{8}\right) = 7 \log x + 8 \log y \] So the simplified expression is: \[ 7 \log x + 8 \log y \]