Express as a single logarithm and simplify, if possible. \[ \frac{1}{4} \log _{e} x+3 \log _{e} y-4 \log _{e} x \]

Step 1 ：Given the expression \(\frac{1}{4} \log _{e} x+3 \log _{e} y-4 \log _{e} x\), we can use the properties of logarithms to simplify.

Step 2 ：First, we can combine the terms involving \(\log _{e} x\): \(\frac{1}{4} \log _{e} x - 4 \log _{e} x = -\frac{15}{4} \log _{e} x\).

Step 3 ：So the expression becomes \(-\frac{15}{4} \log _{e} x + 3 \log _{e} y\).

Step 4 ：Next, we can use the power rule of logarithms, which states that \(a \log_b c = \log_b (c^a)\), to rewrite the expression as \(\log _{e} x^{-\frac{15}{4}} + \log _{e} y^3\).

Step 5 ：Finally, we can use the product rule of logarithms, which states that \(\log_b m + \log_b n = \log_b (mn)\), to combine the two terms into a single logarithm: \(\log _{e} (x^{-\frac{15}{4}}y^3)\).

Step 6 ：Thus, the expression \(\frac{1}{4} \log _{e} x+3 \log _{e} y-4 \log _{e} x\) simplifies to \(\boxed{\log _{e} (x^{-\frac{15}{4}}y^3)}\).