$K$
Express as a sum or difference of logarithms without exponents.
\[
\log _{c} \sqrt[9]{\frac{x^{5}}{y^{9} z^{8}}}
\]

Step 1 ：The given expression is a logarithm with a base of c and an argument that is a ninth root of a fraction. The fraction's numerator is \(x^5\) and the denominator is \(y^9z^8\).

Step 2 ：The first step is to simplify the expression inside the logarithm. The ninth root of a number is the same as raising that number to the power of \(\frac{1}{9}\).

Step 3 ：The second step is to use the properties of logarithms to simplify the expression further. The logarithm of a fraction is equal to the logarithm of the numerator minus the logarithm of the denominator.

Step 4 ：The third step is to use the power rule of logarithms, which states that the logarithm of a number raised to an exponent is equal to the exponent times the logarithm of the number.

Step 5 ：The expression \(\log _{c} \sqrt[9]{\frac{x^{5}}{y^{9} z^{8}}}\) can be simplified to \(\frac{\log _{c}\left(\frac{x^{5}}{y^{9} z^{8}}\right)}{9 \log (c)}\).

Step 6 ：The next step is to apply the properties of logarithms to the simplified expression. The logarithm of a fraction is equal to the logarithm of the numerator minus the logarithm of the denominator. Also, the logarithm of a number raised to an exponent is equal to the exponent times the logarithm of the number.

Step 7 ：The final expression is \(\frac{5 \log _{c}(x)}{9} - \frac{\log _{c}(y)}{9} - \frac{8 \log _{c}(z)}{9}\).

Step 8 ：\(\boxed{\frac{5 \log _{c}(x)}{9} - \frac{\log _{c}(y)}{9} - \frac{8 \log _{c}(z)}{9}}\) is the final answer.