Write the expression as a single logarithm with coefficient 1. Assume all variables represent positive real numbers.
\[
-\frac{2}{3} \log _{6} 6 m^{2}+\frac{1}{2} \log _{6} 36 m^{2}
\]

Step 1 ：Given the expression \(-\frac{2}{3} \log _{6} 6 m^{2}+\frac{1}{2} \log _{6} 36 m^{2}\), we need to write it as a single logarithm with coefficient 1. Assume all variables represent positive real numbers.

Step 2 ：The properties of logarithms state that \(n \log_a b = \log_a b^n\) and \(\log_a b + \log_a c = \log_a (b*c)\). Therefore, we can rewrite the expression as a single logarithm by first applying the power rule to each term, and then applying the product rule to combine the two terms into one.

Step 3 ：Applying the power rule, the expression becomes \(\log _{6} (6 m^{2})^{-\frac{2}{3}}+\log _{6} (36 m^{2})^{\frac{1}{2}}\).

Step 4 ：Then, applying the product rule, the expression becomes \(\log _{6} (6 m^{2})^{-\frac{2}{3}}(36 m^{2})^{\frac{1}{2}}\).

Step 5 ：\(\boxed{\log _{6} (6 m^{2})^{-\frac{2}{3}}(36 m^{2})^{\frac{1}{2}}}\) is the expression written as a single logarithm with coefficient 1.