Решебник по алгебре и начала математического анализа 11 класс Алимов Задание 343

\[\boxed{\mathbf{343}\mathbf{.}}\]

\[1)\log_{5}x^{2} = 0\]

\[\log_{5}x^{2} = \log_{5}1\]

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

\[x = \pm 1.\]

\[имеет\ смысл\ при:\]

\[x^{2} > 0\ \]

\[x \neq 0.\]

\[Ответ:\ \ x = \pm 1.\]

\[2)\log_{4}x^{2} = 3\]

\[\log_{4}x^{2} = \log_{4}4^{3}\]

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

\[x^{2} = 64\]

\[x = \pm 8.\]

\[имеет\ смысл\ при:\]

\[x^{2} > 0\]

\[x \neq 0.\]

\[Ответ:\ \ x = \pm 8.\]

\[3)\log_{3}x^{3} = 0\]

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

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

\[x = 1.\]

\[имеет\ смысл\ при:\]

\[x^{3} > 0\ \]

\[x > 0.\]

\[Ответ:\ \ x = 1.\]

\[4)\log_{4}x^{3} = 6\]

\[\log_{4}x^{3} = \log_{4}4^{6}\]

\[x^{3} = 4^{6}\]

\[x^{3} = \left( 4^{2} \right)^{3}\]

\[x^{3} = 16^{3}\ \]

\[x = 16.\]

\[имеет\ смысл\ при:\]

\[x^{3} > 0\ \]

\[x > 0.\]

\[Ответ:\ \ x = 16.\]

\[5)\lg x^{4} + \lg(4x) = 2 + \lg x^{3}\]

\[\lg x^{4} + \lg(4x) - \lg x^{3} = 2\]

\[\lg\frac{x^{4} \bullet 4x}{x^{3}} = \lg 10^{2}\]

\[\lg{4x^{2}} = \lg 100\]

\[4x^{2} = 100\]

\[x^{2} = 25\]

\[x = \pm 5.\]

\[имеет\ смысл\ при:\]

\[x^{4} > 0 \Longrightarrow x \neq 0;\]

\[4x > 0 \Longrightarrow x > 0;\]

\[x^{3} > 0 \Longrightarrow x > 0;\]

\[Ответ:\ \ x = 5.\]

\[6)\lg x + \lg x^{2} = \lg(9x)\]

\[\lg x + \lg x^{2} - \lg(9x) = 0\]

\[\lg\frac{x \bullet x^{2}}{9x} = 0\]

\[\lg\frac{x^{2}}{9} = 0\]

\[\lg x^{2} - \lg 9 = 0\]

\[\lg x^{2} = \lg 9\]

\[x^{2} = 9\]

\[x = \pm 3.\]

\[имеет\ смысл\ при:\]

\[x > 0;\]

\[x^{2} > 0 \Longrightarrow x \neq 0;\]

\[9x > 0 \Longrightarrow \ x > 0.\]

\[Ответ:\ \ x = 3.\]