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МатематикаРешебник по алгебре и начала математического анализа 10 класс Алимов Задание 1355
Задание 1355
\[\boxed{\mathbf{1355}\mathbf{.}}\]
\[1)\ x^{\lg x} = 10\]
\[\log_{x}x^{\lg x} = \log_{x}10\]
\[\lg x = \frac{\lg 10}{\lg x}\]
\[\lg x = \frac{1}{\lg x}\ \ \ \ \ | \bullet \lg x\]
\[\lg^{2}x = 1\]
\[\lg x = \pm 1.\]
\[1)\ \lg x = - 1\]
\[\lg x = \lg 10^{- 1}\]
\[x = 0,1.\]
\[2)\ \lg x = 1\]
\[\lg x = \lg 10\]
\[x = 10.\]
\[Ответ:\ \ x_{1} = 0,1;\ \ x_{2} = 10.\]
\[2)\ x^{\log_{3}x} = 9x\]
\[\log_{x}x^{\log_{3}x} = \log_{x}{9x}\]
\[\log_{3}x = \log_{x}9 + \log_{x}x\]
\[\log_{3}x = \frac{\log_{3}9}{\log_{3}x} + 1\]
\[\log_{3}x = \frac{2}{\log_{3}x} + 1\ \ \ \ \ | \bullet \log_{3}x\]
\[\left( \log_{3}x \right)^{2} = 2 + \log_{3}x\]
\[y = \log_{3}x:\]
\[y^{2} - y - 2 = 0\]
\[D = 1 + 8 = 9:\]
\[y_{1} = \frac{1 - 3}{2} = - 1;\]
\[y_{2} = \frac{1 + 3}{2} = 2.\]
\[1)\ \log_{3}x = - 1\]
\[\log_{3}x = \log_{3}3^{- 1}\]
\[x = \frac{1}{3}.\]
\[2)\ \log_{3}x = 2\]
\[\log_{3}x = \log_{3}3^{2}\]
\[x = 9.\]
\[Ответ:\ \ x_{1} = \frac{1}{3};\ \ x_{2} = 9.\]
\[3)\ x^{\lg x} - 1 = 10\left( 1 - x^{- \lg x} \right)\]
\[x^{\lg x} - 1 = 10 - 10x^{- \lg x}\]
\[x^{\lg x} + 10x^{- \lg x} - 11 = 0\]
\[y = x^{\lg x}:\]
\[y + 10y^{- 1} - 11 = 0\ \ \ \ \ | \bullet y\]
\[y^{2} - 11y + 10 = 0\]
\[D = 121 - 40 = 81\]
\[y_{1} = \frac{11 - 9}{2} = 1;\]
\[y_{2} = \frac{11 + 9}{2} = 10.\]
\[1)\ x^{\lg x} = 1\]
\[\log_{x}x^{\lg x} = \log_{x}1\]
\[\lg x = 0\]
\[x = 1.\]
\[2)\ x^{\lg x} = 10\]
\[\log_{x}x^{\lg x} = \log_{x}10\]
\[\lg x = \frac{\lg 10}{\lg x}\]
\[\lg x = \frac{1}{\lg x}\ \ \ \ \ | \bullet \lg x\]
\[\lg^{2}x = 1\]
\[\lg x = \pm 1.\]
\[1)\ \lg x = - 1\ \ \]
\[\lg x = \lg 10^{- 1}\]
\[x = 0,1.\]
\[2)\ \lg x = 1\ \]
\[\lg x = \lg 10\]
\[x = 10.\]
\[Ответ:\ \ \]
\[x_{1} = 1;\ \ x_{2} = 0,1;\ \ x_{3} = 10.\]
\[4)\ x^{\sqrt{x}} = \sqrt{x^{x}};\ \ x \neq 0;\ \ x > 0.\]
\[x = 1;\]
\[x^{\sqrt{x}} = x^{\frac{x}{2}}\]
\[\sqrt{x} = \frac{x}{2}\]
\[2\sqrt{x} = x\]
\[4x = x^{2}\]
\[x^{2} - 4x = 0\]
\[x \bullet (x - 4) = 0\]
\[x_{1} = 0\ \ и\ \ x_{2} = 4.\]
\[Ответ:\ \ x_{1} = 1;\ \ x_{2} = 4.\]
