Complete the sentence below. If \( \log _{4} M=\frac{\log _{8} 9}{\log _{8} 4} \), then \( M= \)

To solve for \( M \), we can start by simplifying the expression on the right side of the equation. Using the change of base formula for logarithms, we have: \[ \log_{4} M = \frac{\log_{8} 9}{\log_{8} 4} \] Now we can convert the logarithms with a base of \(8\) to a base of \(4\): \[ \log_{8} 9 = \frac{\log_{4} 9}{\log_{4} 8} \] and \[ \log_{8} 4 = \frac{\log_{4} 4}{\log_{4} 8} \] Since \(\log_{4} 4 = 1\), we can rewrite the equation as: \[ \log_{4} M = \frac{\log_{4} 9}{\frac{1}{\log_{4} 8}} = \log_{4} 9 \cdot \log_{4} 8 \] Next, since \( \log_{4} 8 = \log_{4} (2^3) = 3 \log_{4} 2 \) and \( \log_{4} 9 = \log_{4} (3^2) = 2 \log_{4} 3 \), we will substitute these back into the equation: \[ \log_{4} M = (2 \log_{4} 3)(3 \log_{4} 2) = 6 \log_{4} 3 \log_{4} 2 \] Now, using the identity \( \log_{a} b \cdot \log_{a} c = \log_{a} (b^c) \) on the right side, we get: \[ \log_{4} M = \log_{4} (3^{6}) = \log_{4} 729 \] Since the logarithms are equal, we can conclude that: \[ M = 729 \] So, the final answer is: \[ M = 729 \]