Write as the sum and/or difference of logarithms. Express powers as factors. \( \log _{6}\left(\frac{x y^{6}}{z^{3}}\right) \)

To express \( \log_{6}\left(\frac{x y^{6}}{z^{3}}\right) \) as the sum and/or difference of logarithms, we can utilize logarithm properties. First, apply the quotient rule: \[ \log_{6}\left(\frac{x y^{6}}{z^{3}}\right) = \log_{6}(x y^{6}) - \log_{6}(z^{3}). \] Next, use the product rule on \( \log_{6}(x y^{6}) \): \[ \log_{6}(x y^{6}) = \log_{6}(x) + \log_{6}(y^{6}). \] Now, for \( \log_{6}(y^{6}) \), apply the power rule: \[ \log_{6}(y^{6}) = 6\log_{6}(y). \] Thus, we have: \[ \log_{6}(x y^{6}) = \log_{6}(x) + 6\log_{6}(y). \] Combining all parts together: \[ \log_{6}\left(\frac{x y^{6}}{z^{3}}\right) = \log_{6}(x) + 6\log_{6}(y) - \log_{6}(z^{3}). \] \[ \log_{6}(z^{3}) = 3\log_{6}(z). \] Putting it all together gives us: \[ \log_{6}\left(\frac{x y^{6}}{z^{3}}\right) = \log_{6}(x) + 6\log_{6}(y) - 3\log_{6}(z). \]