Решебник по алгебре и начала математического анализа 11 класс Алимов Задание 1355

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\[1)\ x^{\lg x} = 10\]

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

\[\lg x = \frac{\lg 10}{\lg x}\]

\[\lg x = \frac{1}{\lg x}\ \ \ \ \ | \bullet \lg x\]

\[\lg^{2}x = 1\]

\[\lg x = \pm 1.\]

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

\[\lg x = \lg 10^{- 1}\]

\[x = 0,1.\]

\[2)\ \lg x = 1\]

\[\lg x = \lg 10\]

\[x = 10.\]

\[Ответ:\ \ x_{1} = 0,1;\ \ x_{2} = 10.\]

\[2)\ x^{\log_{3}x} = 9x\]

\[\log_{x}x^{\log_{3}x} = \log_{x}{9x}\]

\[\log_{3}x = \log_{x}9 + \log_{x}x\]

\[\log_{3}x = \frac{\log_{3}9}{\log_{3}x} + 1\]

\[\log_{3}x = \frac{2}{\log_{3}x} + 1\ \ \ \ \ | \bullet \log_{3}x\]

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

\[y = \log_{3}x:\]

\[y^{2} - y - 2 = 0\]

\[D = 1 + 8 = 9:\]

\[y_{1} = \frac{1 - 3}{2} = - 1;\]

\[y_{2} = \frac{1 + 3}{2} = 2.\]

\[1)\ \log_{3}x = - 1\]

\[\log_{3}x = \log_{3}3^{- 1}\]

\[x = \frac{1}{3}.\]

\[2)\ \log_{3}x = 2\]

\[\log_{3}x = \log_{3}3^{2}\]

\[x = 9.\]

\[Ответ:\ \ x_{1} = \frac{1}{3};\ \ x_{2} = 9.\]

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

\[x^{\lg x} - 1 = 10 - 10x^{- \lg x}\]

\[x^{\lg x} + 10x^{- \lg x} - 11 = 0\]

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

\[y + 10y^{- 1} - 11 = 0\ \ \ \ \ | \bullet y\]

\[y^{2} - 11y + 10 = 0\]

\[D = 121 - 40 = 81\]

\[y_{1} = \frac{11 - 9}{2} = 1;\]

\[y_{2} = \frac{11 + 9}{2} = 10.\]

\[1)\ x^{\lg x} = 1\]

\[\log_{x}x^{\lg x} = \log_{x}1\]

\[\lg x = 0\]

\[x = 1.\]

\[2)\ x^{\lg x} = 10\]

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

\[\lg x = \frac{\lg 10}{\lg x}\]

\[\lg x = \frac{1}{\lg x}\ \ \ \ \ | \bullet \lg x\]

\[\lg^{2}x = 1\]

\[\lg x = \pm 1.\]

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

\[\lg x = \lg 10^{- 1}\]

\[x = 0,1.\]

\[2)\ \lg x = 1\ \]

\[\lg x = \lg 10\]

\[x = 10.\]

\[Ответ:\ \ \]

\[x_{1} = 1;\ \ x_{2} = 0,1;\ \ x_{3} = 10.\]

\[4)\ x^{\sqrt{x}} = \sqrt{x^{x}};\ \ x \neq 0;\ \ x > 0.\]

\[x = 1;\]

\[x^{\sqrt{x}} = x^{\frac{x}{2}}\]

\[\sqrt{x} = \frac{x}{2}\]

\[2\sqrt{x} = x\]

\[4x = x^{2}\]

\[x^{2} - 4x = 0\]

\[x \bullet (x - 4) = 0\]

\[x_{1} = 0\ \ и\ \ x_{2} = 4.\]

\[Ответ:\ \ x_{1} = 1;\ \ x_{2} = 4.\]