Решебник по алгебре 11 класс Никольский Параграф 10. Равносильность уравнений на множествах Задание 39

\[\boxed{\mathbf{39.}}\]

\[\textbf{а)}\log_{3}x - \frac{2}{1 + \log_{x}27} =\]

\[= \frac{6}{3 + \log_{3}x}\]

\[x > 0;\]

\[x \neq 1;\]

\[\log_{3}x \neq - 3\]

\[x \neq \frac{1}{27};\]

\[\log_{x}27 \neq - 1\]

\[x \neq \frac{1}{27}.\]

\[M =\]

\[= \left( 0;\frac{1}{27} \right) \cup \left( \frac{1}{27};1 \right) \cup (1; + \infty).\]

\[1 + \log_{x}27 = 1 + \log_{x}3^{3}\ =\]

\[= 1 + 3\log_{x}3 = 1 + \frac{3}{\log_{3}x} =\]

\[= \frac{\log_{3}x + 3}{\log_{3}x};\]

\[\frac{2}{{1 + \log_{x}}27} = 2\ :\frac{\log_{3}x + 3}{\log_{3}x} =\]

\[= \frac{2\log_{3}x}{3 + \log_{3}x};\]

\[\log_{3}x - \frac{2\log_{3}x}{3 + \log_{3}x} = \frac{6}{3 + \log_{3}x}\]

\[\log_{3}x = t:\]

\[t - \frac{2t}{3 + t} = \frac{6}{3 + t}\]

\[\frac{3t + t^{2} - 2t - 6}{3 + t} = 0\]

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

\[t_{1} + t_{2} = - 1;\ \ t_{1} \cdot t_{2} = - 6\]

\[t_{1} = - 3;t_{2} = 2.\]

\[\log_{3}x = 2:\]

\[x = 3^{2}\]

\[x = 9 > 1.\]

\[\log_{3}x = - 3\]

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

\[x = \frac{1}{27}\ (не\ подходит).\]

\[Ответ:x = 9.\]

\[\textbf{б)}\log_{2}x - \frac{8}{2 + \log_{2}x} =\]

\[= \frac{4}{1 + \log_{x}4}\]

\[x > 0;\]

\[x \neq 1.\]

\[2 + \log_{2}x \neq 0\]

\[\log_{2}x \neq - 2\]

\[x \neq \frac{1}{4}.\]

\[1 + \log_{x}4 \neq 0\]

\[\log_{x}4 \neq - 1\]

\[x \neq \frac{1}{4}.\]

\[M = \left( 0;\frac{1}{4} \right) \cup \left( \frac{1}{4};1 \right) \cup (1; + \infty).\]

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

\[= 1 + 2\log_{x}2 = 1 + \frac{2}{\log_{2}x} =\]

\[= \frac{\log_{2}x + 2}{\log_{2}x};\]

\[4\ :\frac{\log_{2}x + 2}{\log_{2}x} = \frac{4\log_{2}x}{2 + \log_{2}x};\]

\[\log_{2}x - \frac{8}{2 + \log_{2}x} = \frac{4\log_{2}x}{2 + \log_{2}x}\]

\[\log_{2}x = t:\]

\[t - \frac{8}{2 + t} = \frac{4t}{2 + t}\]

\[\frac{2t + t^{2} - 8 - 4t}{2 + t} = 0\]

\[t^{2} - 2t - 8 = 0\]

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

\[t_{1} = 1 + 3 = 4;\]

\[t_{2} = 1 - 3 = - 2.\]

\[\log_{2}x = 4:\]

\[x = 4^{2}\]

\[x = 16 > 1.\]

\[\log_{2}x = - 2\]

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

\[x = \frac{1}{4}\ (не\ подходит).\]

\[Ответ:x = 16.\]