Expand the logarithm. Assume when necessary that all logarithmic quantities are defined. \( \ln \left(\frac{\sqrt{z}}{x^{6} y}\right)=\square \)

To expand the logarithm, we can use the properties of logarithms: the logarithm of a quotient is the difference of the logarithms and the logarithm of a power is the exponent times the logarithm of the base. First, apply the quotient rule: \[ \ln \left(\frac{\sqrt{z}}{x^{6} y}\right) = \ln(\sqrt{z}) - \ln(x^{6} y) \] Now, expand \(\ln(x^{6} y)\) using the product rule: \[ \ln(x^{6} y) = \ln(x^{6}) + \ln(y) \] Next, apply the power rule to \(\ln(x^{6})\): \[ \ln(x^{6}) = 6 \ln(x) \] Now, substitute everything back: \[ \ln \left(\frac{\sqrt{z}}{x^{6} y}\right) = \ln(\sqrt{z}) - (6 \ln(x) + \ln(y)) \] Finally, simplifying \(\ln(\sqrt{z})\) gives: \[ \ln(\sqrt{z}) = \frac{1}{2} \ln(z) \] Putting it all together, we have: \[ \ln \left(\frac{\sqrt{z}}{x^{6} y}\right) = \frac{1}{2} \ln(z) - 6 \ln(x) - \ln(y) \] Thus, the complete expanded expression is: \[ \frac{1}{2} \ln(z) - 6 \ln(x) - \ln(y) \]