Решебник по алгебре и начала математического анализа 11 класс Алимов Задание 287

\[\boxed{\mathbf{287}\mathbf{.}}\]

\[1)\ \left( 3^{x} + 2^{x} \right)\left( 3^{x} + 3 \bullet 2^{x} \right) = 8 \bullet 6^{x}\]

\[\left( \frac{3}{2} \right)^{2x} - 4 \bullet \left( \frac{3}{2} \right)^{x} + 3 = 0\]

\[Пусть\ y = \left( \frac{3}{2} \right)^{x}:\]

\[y^{2} - 4y + 3 = 0\]

\[D = 4^{2} - 4 \bullet 3 = 16 - 12 = 4\]

\[y_{1} = \frac{4 - 2}{2} = 1;\text{\ \ }y_{2} = \frac{4 + 2}{2} = 3.\]

\[1)\ \left( \frac{3}{2} \right)^{x} = 1\]

\[\left( \frac{3}{2} \right)^{x} = \left( \frac{3}{2} \right)^{0}\ \]

\[x = 0.\]

\[2)\ \left( \frac{3}{2} \right)^{x} = 3\]

\[\log_{\frac{3}{2}}\left( \frac{3}{2} \right)^{x} = \log_{\frac{3}{2}}3\]

\[x = \log_{1,5}3\]

\[Ответ:\ \ x_{1} = 0;\ \ x_{2} = \log_{1,5}3.\]

\[5 \bullet \left( \frac{3}{5} \right)^{2x} - 7 \bullet \left( \frac{3}{5} \right)^{x} - 6 = 0\]

\[Пусть\ y = \left( \frac{3}{5} \right)^{x}:\]

\[5y^{2} - 7y - 6 = 0\]

\[D = 7^{2} + 4 \bullet 5 \bullet 6 = 49 + 120 =\]

\[= 169\]

\[y_{1} = \frac{7 - 13}{2 \bullet 5} = - \frac{6}{10};\]

\[y_{2} = \frac{7 + 13}{2 \bullet 5} = \frac{20}{10} = 2.\]

\[1)\ \left( \frac{3}{5} \right)^{x} = - \frac{6}{10}\]

\[нет\ корней.\]

\[2)\ \left( \frac{3}{5} \right)^{x} = 2\]

\[\log_{\frac{3}{5}}\left( \frac{3}{5} \right)^{x} = \log_{\frac{3}{5}}2\]

\[x = \log_{0,6}2.\]

\[Ответ:\ \ x = \log_{0,6}2.\]