Решебник по алгебре 11 класс Никольский Параграф 8. Уравнения-следствия Задание 14

\[\boxed{\mathbf{14.}}\]

\[\textbf{а)}\log_{2}\left( x^{2} - 3x \right) = \log_{2}(x - 3)\]

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

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

\[D_{1} = 4 - 3 = 1\]

\[x_{1} = 2 + 1 = 3;\]

\[x_{2} = 2 - 1 = 1.\]

\[Проверим:\]

\[\log_{2}(9 - 9) = \log_{2}(3 - 3)\]

\[x = 3 - не\ корень.\]

\[\log_{2}(1 - 3) = \log_{2}(1 - 3)\]

\[a < 0;\]

\[x = 1 - не\ корень.\]

\[Ответ:нет\ корней.\]

\[\textbf{б)}\log_{4}\left( x^{2} - 5x \right) = \log_{4}(x - 9)\]

\[x^{2} - 5x = x - 9\]

\[x^{2} - 6x + 9 = 0\]

\[(x - 3)^{2} = 0\]

\[x = 3.\]

\[Проверим:\]

\[\log_{4}(9 - 15)\log_{4}(3 - 9)\]

\[a < 0;\]

\[x = 3 - не\ корень.\]

\[Ответ:нет\ корней.\]

\[\textbf{в)}\log_{5}\left( x^{2} + 13x \right) = \log_{5}(9x + 5)\]

\[x^{2} + 13x = 9x + 5\]

\[x^{2} + 4x - 5 = 0\]

\[D_{1} = 4 + 5 = 9\]

\[x_{1} = - 2 + 3 = 1;\]

\[x_{2} = - 2 - 3 = - 5.\]

\[Проверка:\]

\[\log_{5}(1 + 13) = \log_{5}(9 + 5)\]

\[\log_{5}14 = \log_{5}14\]

\[x = 1 - корень.\]

\[\log_{5}(25 - 65) = \log_{5}( - 45 + 5)\]

\[a < 0;\]

\[x = - 5 - не\ корень.\]

\[Ответ:x = 1.\]

\[\textbf{г)}\log_{6}\left( x^{2} - x \right) = \log_{6}(6x - 10)\]

\[x^{2} - x = 6x - 10\]

\[x^{2} - 7x + 10 = 0\]

\[x_{1} + x_{2} = 7;\ \ x_{1} \cdot x_{2} = 10\]

\[x_{1} = 2;\ \ x_{2} = 5.\]

\[Проверка:\]

\[\log_{6}(4 - 2) = \log_{6}(12 - 10)\]

\[\log_{6}2 = \log_{6}2\]

\[x = 2 - корень.\]

\[\log_{6}(25 - 5) = \log_{6}(30 - 10)\]

\[\log_{6}20 = \log_{6}20\]

\[x = 5 - корень.\]

\[Ответ:x = 2;x = 5.\]