Use the properties of logarithms to rewrite the expression. Simplify the result as much as possible. Assume all variables represent positive real numbers. \[ \log _{m} \sqrt{\frac{5 r^{5}}{z^{7}}} \] Choose the correct answer.

Step 1 ：Given the expression \(\log _{m} \sqrt{\frac{5 r^{5}}{z^{7}}}\)

Step 2 ：We can use the properties of logarithms to simplify this expression. The properties we will use are:

Step 3 ：1. The square root of a number is the same as raising that number to the power of 1/2.

Step 4 ：2. The logarithm of a quotient is the difference of the logarithms of the numerator and the denominator.

Step 5 ：3. The logarithm of a product is the sum of the logarithms of the factors.

Step 6 ：4. The logarithm of a power is the product of the power and the logarithm of the base.

Step 7 ：Using these properties, we can simplify the given expression.

Step 8 ：First, we replace the square root with a power of 1/2: \(\log _{m} (\frac{5 r^{5}}{z^{7}})^{1/2}\)

Step 9 ：Next, we express the logarithm of the quotient as the difference of the logarithms: \(\frac{1}{2} (\log _{m} 5 r^{5} - \log _{m} z^{7})\)

Step 10 ：Then, we express the logarithm of the product as the sum of the logarithms: \(\frac{1}{2} (\log _{m} 5 + \log _{m} r^{5} - \log _{m} z^{7})\)

Step 11 ：Finally, we simplify the expression to get the final answer: \(\boxed{\frac{\log _{m} 5+\log _{m} r^{5}-\log _{m} z^{7}}{2}}\)