Use the rule \( \log _{b}(x y)=\log _{b} x+\log _{b} y \) to find \( \log _{4}(16 \cdot 64) \). (1 point) Check answer Remaining Attempts : 3

To find \( \log_{4}(16 \cdot 64) \), we can use the property of logarithms. First, we can express the multiplication inside the logarithm: \[ \log_{4}(16 \cdot 64) = \log_{4}(16) + \log_{4}(64) \] Next, we need to evaluate each logarithm separately. We know that \( 16 = 4^2 \), so: \[ \log_{4}(16) = \log_{4}(4^2) = 2 \] Now, for \( 64 \), we can express it in terms of base 4: \[ 64 = 4^3 \] So we have: \[ \log_{4}(64) = \log_{4}(4^3) = 3 \] Now we combine the results: \[ \log_{4}(16 \cdot 64) = \log_{4}(16) + \log_{4}(64) = 2 + 3 = 5 \] Thus, the final answer is \( \log_{4}(16 \cdot 64) = 5 \).