\[1)\ \log_{27}729 = \log_{27}27^{2} = 2;\]
\[2)\ \log_{9}729 = \log_{9}9^{3} = 3;\]
\[3)\ \log_{\frac{1}{3}}729 = \log_{\frac{1}{3}}3^{6} =\]
\[= \log_{\frac{1}{3}}\left( \frac{1}{3} \right)^{- 6} = - 6.\]