$\begin{array}{l}
e)\,\,{3^{ - 1}}{.4^x} + {3.4^x} = \frac{5}{3}{.2^7}\\
\Leftrightarrow \frac{1}{3}{.4^x} + {3.4^x} = \frac{{{{5.2}^7}}}{3}\\
\Leftrightarrow \frac{{10}}{3}.{\left( {{2^2}} \right)^x} = \frac{{10}}{3}{.2^6}\\
\Leftrightarrow {2^{2x}} = {2^6}\\
\Leftrightarrow 2x = 6\\
\Leftrightarrow x = 3.\\
f)\,\,\,{9^{ - x}}{.27^x} = 243\\
\Leftrightarrow {\left( {{3^2}} \right)^{ - x}}.{\left( {{3^3}} \right)^x} = {3^5}\\
\Leftrightarrow {3^{ - 2x}}{.3^{3x}} = {3^5}\\
\Leftrightarrow {3^{3x - 2x}} = {3^5}\\
\Leftrightarrow {3^x} = {3^5}\\
\Leftrightarrow x = 5.\\
e)\,\,\,{2^{ - 2}}{.2^x} + {2.2^x} = {9.2^6}\\
\Leftrightarrow \frac{1}{{{2^2}}}{.2^x} + {2.2^x} = {9.2^6}\\
\Leftrightarrow \frac{9}{4}{.2^x} = {9.2^6}\\
\Leftrightarrow {2^x} = {4.2^6} = {4^8}\\
\Leftrightarrow x = 8.\\
f)\,\,{3^{ - 2}}{.3^4}{.3^x} = {3^7}\\
\Leftrightarrow {3^{ - 2 + 4 + x}} = {3^7}\\
\Leftrightarrow - 2 + 4 + x = 7\\
\Leftrightarrow x = 5.
\end{array}$
