Решебник по алгебре 11 класс Никольский Параграф 8. Уравнения-следствия Задание 29

\[\boxed{\mathbf{29.}}\]

\[\textbf{а)}\ \frac{2}{\left( \log_{x}5 \right)^{2}} - \log_{5}x = 0\]

\[2\left( \log_{5}x \right)^{2} - \log_{5}x = 0\]

\[\log_{5}x\left( 2\log_{5}x - 1 \right) = 0\]

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

\[x = 1.\]

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

\[x = \sqrt{5}.\]

\[Проверка.\]

\[x = 1:\]

\[\frac{2}{\left( \log_{1}5 \right)^{2}} - не\ существует.\]

\[x = \sqrt{5}:\]

\[\frac{2}{\left( \log_{\sqrt{5}}5 \right)^{2}} - \log_{5}\sqrt{5} = 0\]

\[\frac{2}{\left( 2\log_{\sqrt{5}}\sqrt{5} \right)^{2}} - \frac{1}{2} = 0\]

\[\frac{1}{2} - \frac{1}{2} = 0\]

\[0 = 0.\]

\[Ответ:x = \sqrt{5}.\]

\[\textbf{б)}\ \frac{1}{\left( \log_{x}3 \right)^{2}} + \log_{3}x = 0\]

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

\[\log_{3}x\left( \log_{3}x + 1 \right) = 0\]

\[1)\ \log_{3}x = 0\]

\[x = 1.\]

\[2)\ \log_{3}x + 1 = 0\]

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

\[Проверка.\]

\[x = 1:\]

\[\log_{3}1 - не\ существует.\]

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

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

\[\frac{1}{\left( - \log_{\frac{1}{3}}\frac{1}{3} \right)^{2}} - 1 = 0\]

\[1 = 1.\]

\[Ответ:x = \frac{1}{3}.\]

\[\textbf{в)}\ \frac{2}{\left( \log_{x}4 \right)^{2}} + \log_{4}x = 0\]

\[2\left( \log_{4}x \right)^{2} + \log_{4}x = 0\]

\[\log_{4}x\left( 2\log_{4}x + 1 \right) = 0\]

\[\log_{4}x = 0\]

\[x = 1.\]

\[2\log_{4}x + 1 = 0\]

\[x = 2.\]

\[Проверка.\]

\[x = 1:\]

\[\log_{1}4 - не\ существует.\]

\[x = 2:\]

\[\frac{1}{\left( \log_{2}4 \right)^{2}} - \log_{4}2 = 0\]

\[\frac{2}{4} - \frac{1}{2} = 0\]

\[0 = 0.\]

\[Ответ:x = 2.\]

\[\textbf{г)}\ \frac{1}{\left( 2\log_{x}6 \right)^{2}} - \log_{6}x = 0\]

\[\frac{\left( \log_{6}x \right)^{2}}{4} - \log_{6}x = 0\]

\[\log_{6}x\left( \frac{\log_{6}x}{4} - 1 \right) = 0\]

\[\log_{6}x = 0\]

\[x = 1.\]

\[\frac{\log_{6}x}{4} - 1 = 0\]

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

\[Проверка.\ \]

\[x = 1:\]

\[не\ существует\ \log_{1}6.\]

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

\[\frac{1}{\left( 2\log_{4}6 \right)^{2}} - \log_{6}6^{4} = 0\]

\[\frac{1}{\left( 2 \cdot \frac{1}{4} \right)^{2}} - 4 = 0\]

\[4 - 4 = 0\]

\[4 = 4.\]

\[Ответ:x^{6} = 1296.\]