Use the properties of logarithms to rewrite the expression. Simplify the result as much as possible. Assume all variables represent positive real numbers. \[ \log _{m} \sqrt{\frac{2 r^{5}}{z^{7}}} \] Choose the correct answer. $\log _{m} \sqrt{2}+5 \log _{m} \sqrt{r}-7 \log _{m} \sqrt{z}$ $\frac{1}{2}\left(\log _{m} 2+\log _{m} r^{5}+\log _{m} \frac{1}{z^{7}}\right)$ $\frac{1}{2}\left(\log _{m} 2+5 \log _{m} r-7 \log _{m} z\right)$ This cannot be simplified.

Step 1 ：Rewrite the given expression as \(\log _{m} \left(\frac{2 r^{5}}{z^{7}}\right)^{1/2} = \frac{1}{2} \log _{m} \left(\frac{2 r^{5}}{z^{7}}\right)\)

Step 2 ：Use the property of logarithms that the logarithm of a quotient is the difference of the logarithms to rewrite the expression as \(\frac{1}{2} \left(\log _{m} 2 r^{5} - \log _{m} z^{7}\right)\)

Step 3 ：Use the property of logarithms that the logarithm of a product is the sum of the logarithms to rewrite the expression as \(\frac{1}{2} \left(\log _{m} 2 + \log _{m} r^{5} - \log _{m} z^{7}\right)\)

Step 4 ：Use the property of logarithms that the logarithm of a power is the product of the power and the logarithm to rewrite the expression as \(\frac{1}{2} \left(\log _{m} 2 + 5 \log _{m} r - 7 \log _{m} z\right)\)

Step 5 ：Final Answer: \(\boxed{\frac{1}{2}\left(\log _{m} 2+5 \log _{m} r-7 \log _{m} z\right)}\)