Решебник по алгебре 11 класс Никольский Параграф 8. Уравнения-следствия Задание 15

\[\boxed{\mathbf{15.}}\]

\[\textbf{а)}\log_{3}\left( x^{2} - 2x \right) = 1\]

\[\log_{3}\left( x^{2} - 2x \right) = \log_{3}3^{1}\]

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

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

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

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

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

\[Проверка:\]

\[\log_{3}(9 - 6) = 1\]

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

\[1 = 1\]

\[x = 3 - корень.\]

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

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

\[1 = 1\]

\[x = - 1 - корень.\]

\[Ответ:x = - 1;x = 3.\]

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

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

\[x^{2} + 2x = 8\]

\[x^{2} + 2x - 8 = 0\]

\[D_{1} = 1 + 8 = 9\]

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

\[x_{2} = - 1 - 3 = - 4.\]

\[Проверка:\]

\[\log_{2}(4 + 4) = 3\]

\[\log_{2}8 = 3\]

\[3 = 3\]

\[x = 2 - корень.\]

\[\log_{2}(16 - 8) = 3\]

\[\log_{2}8 = 3\]

\[3 = 3\]

\[x = - 4 - корень.\]

\[Ответ:x = - 4;x = 2.\]

\[\textbf{в)}\log_{7}\left( x^{2} + 1,5x \right) = 0\]

\[\log_{7}\left( x^{2} + 1,5x \right) = \log_{7}7^{0}\]

\[x^{2} + 1,5x = 1\]

\[x^{2} + 1,5x - 1 = 0\ \ \ | \cdot 2\]

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

\[D = 9 + 16 = 25\]

\[x_{1} = \frac{- 3 + 5}{4} = \frac{1}{2};\]

\[x_{2} = \frac{- 3 - 5}{4} = - 2.\]

\[Проверка:\]

\[\log_{7}\left( \frac{1}{4} + \frac{3}{2} \cdot \frac{1}{2} \right) = 0\]

\[\log_{7}1 = 0\]

\[0 = 0\]

\[x = \frac{1}{2} - корень.\]

\[\log_{7}(4 - 3) = 0\]

\[\log_{7}1 = 0\]

\[0 = 0\]

\[x = - 2 - корень.\]

\[Ответ:x = - 2;\ \ x = \frac{1}{2}.\]

\[\textbf{г)}\log_{5}\left( x^{2} + 2\frac{2}{3}x \right) = 0\ \]

\[\log_{5}\left( x^{2} + \frac{8}{3}x \right) = \log_{5}5^{0}\]

\[x^{2} + \frac{8}{3}x = 1\]

\[x^{2} + \frac{8}{3}x - 1 = 0\ \ \ | \cdot 3\]

\[3x^{2} + 8x - 3 = 0\]

\[D_{1} = 16 + 9 = 25\]

\[x_{1} = \frac{- 4 + 5}{3} = \frac{1}{3};\]

\[x_{2} = \frac{- 4 - 5}{3} = - 3.\]

\[Проверка:\]

\[\log_{5}\left( \frac{1}{9} + \frac{8}{9} \right) = 0\]

\[\log_{5}1 = 0\]

\[0 = 0\]

\[x = \frac{1}{3} - корень.\]

\[\log_{5}\left( 9 - \frac{8}{3} \cdot 3 \right) = 0\]

\[\log_{5}1 = 0\]

\[0 = 0\]

\[x = - 3 - корень.\]

\[Ответ:x = - 3;x = \frac{1}{3}.\]