\( \log _{2} x+2 \log _{4} y=4 \) sho.0 that \( x y=16 \)

To prove that \( x y = 16 \) given the equation \[ \log_{2} x + 2 \log_{4} y = 4, \] follow these steps: ### Step 1: Express All Logarithms with the Same Base Notice that the logarithms have different bases (2 and 4). To simplify, we'll express the logarithm with base 4 in terms of base 2. Recall that: \[ \log_{4} y = \frac{\log_{2} y}{\log_{2} 4} = \frac{\log_{2} y}{2} \] because \(\log_{2} 4 = 2\). ### Step 2: Substitute Back into the Original Equation Replace \(\log_{4} y\) in the original equation: \[ \log_{2} x + 2 \left( \frac{\log_{2} y}{2} \right) = 4 \] Simplify the equation: \[ \log_{2} x + \log_{2} y = 4 \] ### Step 3: Combine the Logarithms Use the logarithm property that \(\log_{a} m + \log_{a} n = \log_{a} (m \cdot n)\): \[ \log_{2} (x y) = 4 \] ### Step 4: Exponentiate to Remove the Logarithm Convert the logarithmic equation to its exponential form: \[ x y = 2^{4} \] \[ x y = 16 \] ### Conclusion Thus, we have shown that \( x y = 16 \) based on the given logarithmic equation.