Expand. Simplify if possible. Assume that all variables represent positive real numbers. \[ \log _{4}\left(\frac{x y}{z t}\right) \] What is the expanded expression?

Step 1 ：Given the expression \(\log _{4}\left(\frac{x y}{z t}\right)\)

Step 2 ：Using the properties of logarithms, we know that the logarithm of a quotient is the difference of the logarithms of the numerator and the denominator. Also, the logarithm of a product is the sum of the logarithms of the factors.

Step 3 ：Applying these properties, we can expand the given expression as follows: \(\log(x*y/(t*z))/\log(4)\)

Step 4 ：This simplifies to: \(-\log(t)/(2*\log(2)) + \log(x)/(2*\log(2)) + \log(y)/(2*\log(2)) - \log(z)/(2*\log(2))\)

Step 5 ：Finally, we can write the expanded expression as: \(\frac{\log(x)}{2\log(2)} + \frac{\log(y)}{2\log(2)} - \frac{\log(z)}{2\log(2)} - \frac{\log(t)}{2\log(2)}\)

Step 6 ：\(\boxed{\frac{\log(x)}{2\log(2)} + \frac{\log(y)}{2\log(2)} - \frac{\log(z)}{2\log(2)} - \frac{\log(t)}{2\log(2)}}\) is the final answer.