1. **State the problem:** We need to compute $\log_{\frac{1}{8}} 6$ using the change of base formula and round the answer to the nearest thousandth. 2. **Recall the change of base formula:** For any positive numbers $a$, $b$, and base $c$ (with $a \neq 1$, $c \neq 1$), $$\log_a b = \frac{\log_c b}{\log_c a}$$ We can use common logarithms (base 10) or natural logarithms (base $e$). 3. **Apply the formula:** $$\log_{\frac{1}{8}} 6 = \frac{\log 6}{\log \frac{1}{8}}$$ 4. **Calculate the logarithms:** - $\log 6 \approx 0.778151$ (base 10) - $\log \frac{1}{8} = \log 8^{-1} = -\log 8 \approx -0.903090$ 5. **Divide:** $$\frac{0.778151}{-0.903090} \approx -0.861$$ 6. **Interpretation:** The value of $\log_{\frac{1}{8}} 6$ rounded to the nearest thousandth is $-0.861$. **Final answer:** $$\boxed{-0.861}$$