se the properties of logarithms to rewrite the following expression. Assume all variables represent positive real numbers. \( 6 \log _{6} x-7 \log _{6} y \) \( 6 \log _{6} x-7 \log _{6} y=\square \)

To rewrite the expression \( 6 \log_{6} x - 7 \log_{6} y \) using the properties of logarithms, we can apply the following properties: 1. \( a \log_b c = \log_b(c^a) \) (the power rule) 2. \( \log_b a - \log_b c = \log_b\left(\frac{a}{c}\right) \) (the quotient rule) Using these properties, we can rewrite the expression step by step: 1. Apply the power rule to each term: \[ 6 \log_{6} x = \log_{6}(x^6) \] \[ 7 \log_{6} y = \log_{6}(y^7) \] 2. Substitute these into the original expression: \[ 6 \log_{6} x - 7 \log_{6} y = \log_{6}(x^6) - \log_{6}(y^7) \] 3. Now apply the quotient rule: \[ \log_{6}(x^6) - \log_{6}(y^7) = \log_{6}\left(\frac{x^6}{y^7}\right) \] Thus, the expression can be rewritten as: \[ 6 \log_{6} x - 7 \log_{6} y = \log_{6}\left(\frac{x^6}{y^7}\right) \] So, the final answer is: \[ \log_{6}\left(\frac{x^6}{y^7}\right) \]