Write the expression as a single logarithm. \[ \frac{1}{5} \log _{8} z+6 \log _{8} x-\log _{8} y \]

Step 1 ：Given the expression \(\frac{1}{5} \log _{8} z+6 \log _{8} x-\log _{8} y\)

Step 2 ：We can use the properties of logarithms to simplify this expression. The properties are as follows:

Step 3 ：1. \(a \log_b c = \log_b (c^a)\)

Step 4 ：2. \(\log_b a - \log_b c = \log_b \left(\frac{a}{c}\right)\)

Step 5 ：3. \(\log_b a + \log_b c = \log_b (ac)\)

Step 6 ：Applying these properties, the given expression can be simplified as follows:

Step 7 ：\(\frac{1}{5} \log _{8} z+6 \log _{8} x-\log _{8} y = \log _{8} (z^{1/5}) + \log _{8} (x^6) - \log _{8} y\)

Step 8 ：This can be further simplified using the properties 2 and 3 to:

Step 9 ：\(\log _{8} \left(\frac{z^{1/5} x^6}{y}\right)\)

Step 10 ：Final Answer: The expression can be written as a single logarithm as follows:

Step 11 ：\(\boxed{\log _{8} \left(\frac{z^{1/5} x^6}{y}\right)}\)