Use properties of logarithms to expand the logarithmic expression as much as possible. Evaluate logarithmic expressions without using a calculator if possible. \( \log _{4}\left(\frac{16}{\sqrt{x+7}}\right) \) \( \log _{4}\left(\frac{16}{\sqrt{x+7}}\right)=\square \)

To expand \( \log _{4}\left(\frac{16}{\sqrt{x+7}}\right) \), we can use the properties of logarithms: the quotient rule, which states that \( \log_b\left(\frac{A}{B}\right) = \log_b(A) - \log_b(B) \), and the power rule, which says that \( \log_b(A^n) = n \cdot \log_b(A) \). First, we apply the quotient rule: \[ \log_{4}\left(\frac{16}{\sqrt{x+7}}\right) = \log_{4}(16) - \log_{4}(\sqrt{x+7}) \] Now, \( \log_{4}(16) \) can be evaluated since \( 16 = 4^2 \): \[ \log_{4}(16) = \log_{4}(4^2) = 2 \] Next, we simplify \( \log_{4}(\sqrt{x+7}) \): \[ \log_{4}(\sqrt{x+7}) = \log_{4}((x+7)^{1/2}) = \frac{1}{2} \log_{4}(x+7) \] Putting it all together, we have: \[ \log_{4}\left(\frac{16}{\sqrt{x+7}}\right) = 2 - \frac{1}{2} \log_{4}(x+7) \] Thus, the expanded form is: \[ \log_{4}\left(\frac{16}{\sqrt{x+7}}\right) = 2 - \frac{1}{2} \log_{4}(x+7) \] In concise terms, the final answer is: \[ \log_{4}\left(\frac{16}{\sqrt{x+7}}\right) = 2 - \frac{1}{2} \log_{4}(x+7) \]