Use properties of logarithms to expand the logarithmic expression as much as possible. Evaluate loganithmic expressions without using a calculator if possible. \( \log _{4}\left(\frac{16}{\sqrt{x+9}}\right) \) \( \log _{4}\left(\frac{16}{\sqrt{x+9}}\right)=\square \)

To expand the logarithmic expression, we can utilize several properties of logarithms. First, we will apply the quotient rule which states that \( \log_b\left(\frac{m}{n}\right) = \log_b(m) - \log_b(n) \). Thus, we can write: \[ \log_{4}\left(\frac{16}{\sqrt{x+9}}\right) = \log_{4}(16) - \log_{4}(\sqrt{x+9}). \] Next, we can evaluate \( \log_{4}(16) \). Since \( 16 = 4^2 \), it follows that \( \log_{4}(16) = 2 \). For \( \log_{4}(\sqrt{x+9}) \), we know that \( \sqrt{x+9} = (x+9)^{1/2} \). Using the power rule \( \log_b(m^n) = n \cdot \log_b(m) \), we have: \[ \log_{4}(\sqrt{x+9}) = \frac{1}{2}\log_{4}(x+9). \] Putting this all together, we combine everything: \[ \log_{4}\left(\frac{16}{\sqrt{x+9}}\right) = 2 - \frac{1}{2}\log_{4}(x+9). \] So the final expanded expression is: \[ \boxed{2 - \frac{1}{2}\log_{4}(x+9)}. \]