Решебник по алгебре и начала математического анализа 11 класс Колягин Задание 945

\[1)\ \left\{ \begin{matrix} 2^{x + y} = 32\ \ \\ 3^{3y - x} = 27 \\ \end{matrix} \right.\ \]

\[2^{x + y} = 32\]

\[2^{x + y} = 2^{5}\]

\[x + y = 5\]

\[y = 5 - x.\]

\[3^{3y - x} = 27\]

\[3^{3(5 - x) - x} = 3^{3}\]

\[15 - 3x - x = 3\]

\[4x = 12\]

\[x = 3;\]

\[y = 5 - 3 = 2.\]

\[Ответ:\ \ (3;\ 2).\]

\[2)\ \left\{ \begin{matrix} 3^{x} - 2 \bullet 2^{y} = 77 \\ 3^{\frac{x}{2}} - 2^{y} = 7\ \ \ \ \ \ \ \ \\ \end{matrix} \right.\ \]

\[3^{\frac{x}{2}} = 2^{y} + 7\]

\[3^{x} = 2^{2y} + 14 \bullet 2^{y} + 49.\]

\[3^{x} - 2 \bullet 2^{y} = 77\]

\[2^{2y} + 14 \bullet 2^{y} + 49 - 2 \bullet 2^{y} = 77\]

\[2^{2y} + 12 \bullet 2^{y} - 28 = 0\]

\[D = 144 + 112 = 256\]

\[2_{1}^{y} = \frac{- 12 - 16}{2} = - 14;\]

\[2_{2}^{y} = \frac{- 12 + 16}{2} = 2;\]

\[2^{y} = 2\]

\[y = 1.\]

\[3^{x} = 2^{2} + 14 \bullet 2^{1} + 49\]

\[3^{x} = 4 + 28 + 49\]

\[3^{x} = 81\]

\[3^{x} = 3^{4}\]

\[x = 4.\]

\[Ответ:\ \ (4;\ 1).\]

\[3)\ \left\{ \begin{matrix} 3^{x} \bullet 2^{y} = 576\ \ \ \ \ \ \ \\ \log_{\sqrt{2}}(y - x) = 4 \\ \end{matrix} \right.\ \]

\[\log_{\sqrt{2}}(y - x) = 4\]

\[y - x = \left( \sqrt{2} \right)^{4}\]

\[y - x = 4\]

\[y = 4 + x.\]

\[3^{x} \bullet 2^{y} = 576.\]

\[3^{x} \bullet 2^{4 + x} = 9 \bullet 64\]

\[3^{x} \bullet 2^{4 + x} = 3^{2} \bullet 2^{4 + 2}\]

\[x = 2;\]

\[y = 4 + 2 = 6.\]

\[Ответ:\ \ (2;\ 6).\]

\[4)\ \left\{ \begin{matrix} \lg x + \lg y = 4 \\ x^{\lg y} = 1000\ \ \ \\ \end{matrix} \right.\ \]

\[\lg x + \lg y = 4\]

\[\lg\text{xy} = 4\]

\[xy = 10^{4}\]

\[y = \frac{10^{4}}{x}.\]

\[x^{\lg y} = 1000\]

\[\log_{x}x^{\lg y} = \log_{x}10^{3}\]

\[\lg y = \log_{x}10^{3}\]

\[\lg\frac{10^{4}}{x} = \log_{x}10^{3}\]

\[\lg 10^{4} - \lg x = \frac{\lg 10^{3}}{\lg x}\]

\[4 - \lg x = \frac{3}{\lg x}\ \ \ \ \ | \bullet \lg x\]

\[4\lg x - \lg^{2}x = 3.\]

\[\ a = \lg x:\]

\[4a - a^{2} = 3\]

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

\[D = 16 - 12 = 4\]

\[a_{1} = \frac{4 - 2}{2} = 1;\]

\[a = \frac{4 + 2}{2} = 3.\]

\[1)\ \lg x = 1\]

\[x = 10;\]

\[y = \frac{10^{4}}{10} = 10^{3}.\]

\[2)\ \lg x = 3\]

\[x = 10^{3};\]

\[y = \frac{10^{4}}{10^{3}} = 10.\]

\[Ответ:\ \ (10;\ 1000);\ (1000;\ 10).\]