Use the properties of logarithms to expand the logarithm. Simplify if possible. \( \log _{6}\left(\sqrt[5]{\frac{7 x^{5}}{3 y^{6} z^{4}}}\right) \)

To expand \( \log _{6}\left(\sqrt[5]{\frac{7 x^{5}}{3 y^{6} z^{4}}}\right) \), we first rewrite the expression using the property of logarithms for roots. The square root can be written as a power, so we have: \[ \log _{6}\left(\sqrt[5]{\frac{7 x^5}{3 y^6 z^4}}\right) = \frac{1}{5} \log _{6}\left(\frac{7 x^{5}}{3 y^{6} z^{4}}\right) \] Next, we can apply the properties of logarithms to expand the logarithm of a quotient: \[ = \frac{1}{5} \left( \log _{6}(7 x^{5}) - \log _{6}(3 y^{6} z^{4}) \right) \] We can further expand both logarithms using the product property: \[ = \frac{1}{5} \left( \log _{6}(7) + \log _{6}(x^{5}) - (\log _{6}(3) + \log _{6}(y^{6}) + \log _{6}(z^{4})) \right) \] Now, applying the power property on \( x^5 \), \( y^6 \), and \( z^4 \): \[ = \frac{1}{5} \left( \log _{6}(7) + 5 \log _{6}(x) - \log _{6}(3) - 6 \log _{6}(y) - 4 \log _{6}(z) \right) \] Finally, we can distribute the \( \frac{1}{5} \): \[ = \frac{1}{5} \log _{6}(7) + \log _{6}(x) - \frac{1}{5} \log _{6}(3) - \frac{6}{5} \log _{6}(y) - \frac{4}{5} \log _{6}(z) \] So, the final expanded and simplified form is: \[ \boxed{\frac{1}{5} \log_{6}(7) + \log_{6}(x) - \frac{1}{5} \log_{6}(3) - \frac{6}{5} \log_{6}(y) - \frac{4}{5} \log_{6}(z)} \]