Решебник по алгебре и начала математического анализа 10 класс Алимов Задание 1573

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\[\left\{ \begin{matrix} 3^{\log_{x}2} = y^{\log_{5}y} \\ 2^{\log_{y}3} = x^{\log_{7}x} \\ \end{matrix} \right.\ \]

\[\left\{ \begin{matrix} \log_{3}3^{\log_{x}2} = \log_{3}y^{\log_{5}y} \\ \log_{2}2^{\log_{y}3} = \log_{2}x^{\log_{7}x} \\ \end{matrix} \right.\ \]

\[\left\{ \begin{matrix} \log_{x}2 = \log_{3}y^{\log_{5}y} \\ \log_{y}3 = \log_{2}x^{\log_{7}x} \\ \end{matrix} \right.\ \]

\[\left\{ \begin{matrix} \frac{\log_{2}2}{\log_{2}x} = \log_{5}y \bullet \log_{3}y \\ \frac{\log_{3}3}{\log_{3}y} = \log_{7}x \bullet \log_{2}x \\ \end{matrix} \right.\ \]

\[\left\{ \begin{matrix} \frac{1}{\log_{2}x} = \log_{5}y \bullet \log_{3}y \\ \frac{1}{\log_{3}y} = \log_{7}x \bullet \log_{2}x \\ \end{matrix} \right.\ \]

\[Первое\ уравнение:\]

\[\log_{2}x = \frac{1}{\log_{5}y \bullet \log_{3}y}\]

\[\log_{2}x = \log_{2}2^{\frac{1}{\log_{5}y \bullet \log_{3}y}}\]

\[x = 2^{\frac{1}{\log_{5}y \bullet \log_{3}y}}.\]

\[Второе\ уравнение:\]

\[\frac{1}{\log_{3}y} = \log_{7}2^{\frac{1}{\log_{5}y \bullet \log_{3}y}} \bullet \log_{2}2^{\frac{1}{\log_{5}y \bullet \log_{3}y}}\]

\[\frac{1}{\log_{3}y} = \frac{\log_{2}2^{\frac{1}{\log_{5}y \bullet \log_{3}y}}}{\log_{2}7} \bullet \frac{1}{\log_{5}y \bullet \log_{3}y}\]

\[\frac{1}{\log_{3}y} = \frac{1}{\log_{5}^{2}y \bullet \log_{3}^{2}y \bullet \log_{2}7}\]

\[\log_{3}y = \log_{5}^{2}y \bullet \log_{3}^{2}y \bullet \log_{2}7\]

\[\log_{5}^{2}y \bullet \log_{3}y = \frac{1}{\log_{2}7}\]

\[\log_{5}^{2}y \bullet \frac{\log_{5}y}{\log_{5}3} = \log_{7}2\]

\[\log_{5}^{3}y = \log_{5}3 \bullet \log_{7}2\]

\[y = 5^{\left( \log_{5}3 \bullet \log_{7}2 \right)^{\frac{1}{3}}}.\]

\[\frac{1}{\log_{2}x} = \log_{5}5^{\left( \log_{5}3 \bullet \log_{7}2 \right)^{\frac{1}{3}}} \bullet \log_{3}5^{\left( \log_{5}3 \bullet \log_{7}2 \right)^{\frac{1}{3}}}\]

\[\frac{1}{\log_{2}x} = \left( \log_{5}3 \bullet \log_{7}2 \right)^{\frac{1}{3}} \bullet \frac{\log_{5}5^{\left( \log_{5}3 \bullet \log_{7}2 \right)^{\frac{1}{3}}}}{\log_{5}3}\]

\[\frac{1}{\log_{2}x} = \frac{\left( \log_{5}3 \bullet \log_{7}2 \right)^{\frac{2}{3}}}{\log_{5}3}\]

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

\[x = 2^{\frac{\log_{5}3}{\left( \log_{5}3 \bullet \log_{7}2 \right)^{\frac{2}{3}}}}.\]

\[Ответ:\ \ \]

\[\left( 2^{\frac{\log_{5}3}{\left( \log_{5}3 \bullet \log_{7}2 \right)^{\frac{2}{3}}}};\ \ 5^{\left( \log_{5}3 \bullet \log_{7}2 \right)^{\frac{1}{3}}} \right).\]