The equation is \(\log_m m + \log_{\frac{1}{6}} \frac{1}{3} = 2\). Because \(\log_m m = 1\), we simplify \(\log_{\frac{1}{6}} \frac{1}{3}\) using the change of base formula:
\[\n\log_{\frac{1}{6}} \frac{1}{3} = \frac{\log \frac{1}{3}}{\log \frac{1}{6}}\n\]
This leads to:
\[\nm = 12\n\]
The answer is \(12\).