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

Step 1 ：Given the expression ${\log }_m\sqrt{\frac{5r^5}{z^7}}$

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: ${\log }_m(\frac{5r^5}{z^7}{)}^{1/2}$

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

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: $\boxed{{\displaystyle \frac{{\log }_{m}5+{\log }_m{r}^5-{\log }_m{z}^7}{2}}}$