Step 1 :Given the expression \(\log _{8}\left(\frac{\sqrt{9}}{11}\right)\)
Step 2 :According to the properties of logarithms, the logarithm of a quotient is equal to the difference of the logarithms of the numerator and the denominator. So, we can rewrite the expression as \(\log _{8}(\sqrt{9}) - \log _{8}(11)\)
Step 3 :Also, the square root of a number is the same as raising that number to the power of 1/2. So, we can further rewrite the expression as \(\frac{1}{2} \log _{8}(9) - \log _{8}(11)\)
Step 4 :Final Answer: \(\boxed{\text{D. } \frac{1}{2} \log _{8} 9-\log _{8} 11}\)