Use the change of base formula to compute $\log _{9} \frac{1}{7}$. Round your answer to the nearest thousandth.

Step 1 ：Given the problem to compute \(\log _{9} \frac{1}{7}\), we can use the change of base formula.

Step 2 ：The change of base formula is given by \(\log_b a = \frac{\log_c a}{\log_c b}\) for any positive base \(c\) different from 1.

Step 3 ：We can use this formula to compute \(\log _{9} \frac{1}{7}\) by choosing \(c\) to be 10, which is the base of the common logarithm.

Step 4 ：This gives us \(\log _{9} \frac{1}{7} = \frac{\log_{10} \frac{1}{7}}{\log_{10} 9}\).

Step 5 ：Computing the numerator \(\log_{10} \frac{1}{7}\), we get -0.8450980400142569.

Step 6 ：Computing the denominator \(\log_{10} 9\), we get 0.9542425094393249.

Step 7 ：Dividing the numerator by the denominator, we get -0.8856218745807112.

Step 8 ：Rounding this to the nearest thousandth, we get -0.886.

Step 9 ：Final Answer: \(\boxed{-0.886}\)