Which expression is equivalent to \( \log _{4} x^{7} \) ?

To simplify \( \log_{4} x^{7} \), we can utilize the power rule of logarithms, which states that \( \log_{b} a^{c} = c \cdot \log_{b} a \). Applying this rule, we get: \[ \log_{4} x^{7} = 7 \cdot \log_{4} x \] Thus, the expression \( \log_{4} x^{7} \) is equivalent to \( 7 \log_{4} x \). Now, if you want to express it in terms of base 2, a clever trick is to remember that \( 4 = 2^{2} \), and use the change of base formula: \[ \log_{4} x = \frac{\log_{2} x}{\log_{2} 4} = \frac{\log_{2} x}{2} \] So that expands \( \log_{4} x^{7} \) to: \[ 7 \cdot \frac{\log_{2} x}{2} = \frac{7}{2} \log_{2} x \]