Expand the logarithm. Assume when necessary that all logarithmic quantities are defined. \( \log _{2}\left(\frac{x}{9}\right)^{4}=\square \)

To expand the logarithm \( \log_{2}\left(\frac{x}{9}\right)^{4} \), we can use the power rule of logarithms, which states that \( \log_{b}(a^{n}) = n \cdot \log_{b}(a) \). Thus, we can write: \[ \log_{2}\left(\frac{x}{9}\right)^{4} = 4 \cdot \log_{2}\left(\frac{x}{9}\right) \] Next, we will apply the quotient rule of logarithms, which states that \( \log_{b}\left(\frac{a}{c}\right) = \log_{b}(a) - \log_{b}(c) \). So, it becomes: \[ 4 \cdot \log_{2}\left(\frac{x}{9}\right) = 4 \left(\log_{2}(x) - \log_{2}(9)\right) \] Now, combining these results, we expand it fully to get: \[ \log_{2}\left(\frac{x}{9}\right)^{4} = 4\log_{2}(x) - 4\log_{2}(9) \] Final expansion: \[ \log_{2}\left(\frac{x}{9}\right)^{4} = 4\log_{2}(x) - 4\log_{2}(9) \]