Write the expression as a single logarithm.
\[
2\left(\log _{4} w-4 \log _{4} y\right)+2 \log _{4} z
\]
Answer
The final answer is \(\boxed{\log _{4}\left(\frac{w^{2} z^{2}}{y^{8}}\right)}\)
Step 1 ：The given expression is \(2\left(\log _{4} w-4 \log _{4} y\right)+2 \log _{4} z\)
Step 2 ：We can simplify it by using the properties of logarithms. The properties we will use are: \(\log_b(a^n) = n * \log_b(a)\), \(\log_b(a) - \log_b(b) = \log_b(a/b)\), and \(\log_b(a) + \log_b(b) = \log_b(ab)\)
Step 3 ：Applying these properties to the given expression, we get \(\log(w^2)/\log(4) - 8*\log(y)/\log(4) + 2*\log(z)/\log(4)\)
Step 4 ：Simplifying further, we get \(\log(w^2*z^2/y^8)/\log(4)\)
Step 5 ：The final answer is \(\boxed{\log _{4}\left(\frac{w^{2} z^{2}}{y^{8}}\right)}\)
