Write the expression as a single logarithm with coefficient 1 . Assume that all variables represent positive. real numbers. \[ 4 \log _{c} x-6 \log _{c} y^{5} \]

Step 1 ：Given the expression \(4 \log _{c} x-6 \log _{c} y^{5}\)

Step 2 ：Using the properties of logarithms, we can move the coefficient of each logarithm as the exponent of the argument inside the logarithm. This gives us \(\log _{c} x^{4} - \log _{c} (y^{5})^{6}\)

Step 3 ：Next, we write the difference of two logarithms as the logarithm of the quotient of the arguments of the two logarithms. This gives us \(\log _{c} \frac{x^{4}}{(y^{5})^{6}}\)

Step 4 ：Thus, the expression \(4 \log _{c} x-6 \log _{c} y^{5}\) simplifies to \(\log _{c} \frac{x^{4}}{(y^{5})^{6}}\)

Step 5 ：Final Answer: \(\boxed{\log _{c} \frac{x^{4}}{(y^{5})^{6}}}\)