Write the logarithm as a sum and/or difference of logarithmic expressions. Eliminate exponents and radicals, and evaluate logarithms wherever possible. Assume that \( x, y, z>0 \). \( \ln \left(\frac{\sqrt[3]{x y^{3}}}{z^{5}}\right)= \)

Let's start by breaking down the expression using the properties of logarithms! First, we can apply the quotient rule of logarithms, which states that \( \ln \left(\frac{A}{B}\right) = \ln(A) - \ln(B) \). Thus, we have: \[ \ln \left(\frac{\sqrt[3]{x y^{3}}}{z^{5}}\right) = \ln(\sqrt[3]{x y^{3}}) - \ln(z^{5}). \] Next, we can deal with the logarithm of the individual components. For the first part, we remember that the cube root can be rewritten using exponents: \( \sqrt[3]{A} = A^{1/3} \). Therefore, we can express it as: \[ \ln(\sqrt[3]{x y^{3}}) = \ln((x y^{3})^{1/3}) = \frac{1}{3} \ln(x y^{3}). \] Applying the product rule next, \( \ln(AB) = \ln(A) + \ln(B) \), gives us: \[ \frac{1}{3} \ln(x y^{3}) = \frac{1}{3} (\ln(x) + \ln(y^{3})) = \frac{1}{3} \ln(x) + \frac{1}{3} \cdot 3 \ln(y) = \frac{1}{3} \ln(x) + \ln(y). \] Now, let's simplify the second part \( \ln(z^{5}) \): \[ \ln(z^{5}) = 5 \ln(z). \] Putting it all together, we have: \[ \ln \left(\frac{\sqrt[3]{x y^{3}}}{z^{5}}\right) = \left(\frac{1}{3} \ln(x) + \ln(y)\right) - 5 \ln(z). \] Final expression: \[ \ln \left(\frac{\sqrt[3]{x y^{3}}}{z^{5}}\right) = \frac{1}{3} \ln(x) + \ln(y) - 5 \ln(z). \]