Use the properties of logarithms to rewrite the expression. Simplify the result if possible. Assume all variables represent positive real numbers.
\[
\log _{8}\left(\frac{\sqrt{9}}{11}\right)
\]
A. $\log _{8} \sqrt{9}+\log _{8} 11$
B. $\sqrt{\log _{8} 9}-\log _{8} 11$
C. $\frac{\frac{1}{2} \log _{8} 9}{\log _{8} 11}$
D. $\frac{1}{2} \log _{8} 9-\log _{8} 11$
Answer
Final Answer: \(\boxed{\text{D. } \frac{1}{2} \log _{8} 9-\log _{8} 11}\)
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: \(\boxed{\text{D. } \frac{1}{2} \log _{8} 9-\log _{8} 11}\)
