Решебник по алгебре 11 класс Никольский Параграф 8. Уравнения-следствия Задание 17

\[\boxed{\mathbf{17.}}\]

\[\textbf{а)}\log_{3}{(2 \cdot 3^{x} - 5)} = \log_{3}\left( 3^{x} + 4 \right)\]

\[2 \cdot 3^{x} - 5 = 3^{x} + 4\]

\[3^{x} = 9\]

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

\[x = 2.\]

\[Проверка:\]

\[\log_{3}\left( 2 \cdot 3^{2} - 5 \right) = \log_{3}\left( 3^{2} + 4 \right)\]

\[\log_{3}13 = \log_{3}13\]

\[x = 2 - корень.\]

\[Ответ:x = 2.\]

\[\textbf{б)}\log_{7}\left( 2 \cdot 4^{x} - 3 \right) = \log_{7}\left( 4^{x} + 1 \right)\]

\[2 \cdot 4^{x} - 3 = 4^{x} + 1\]

\[4^{x} = 4\]

\[x = 1.\]

\[Проверка:\]

\[\log_{7}(2 \cdot 4 - 3) = \log_{7}{(4 + 1)}\]

\[\log_{7}5 = \log_{7}5\]

\[x = 1 - корень.\]

\[Ответ:x = 1.\]

\[\textbf{в)}\log_{5}\left( 4^{x} - 3 \cdot 2^{x} \right) =\]

\[= \log_{5}\left( 3 \cdot 2^{x} - 8 \right)\]

\[4^{x} - 3 \cdot 2^{x} = 3 \cdot 2^{x} - 8\]

\[\left( 2^{x} \right)^{2} - 6 \cdot 2^{x} + 8 = 0\]

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

\[t^{2} - 6t + 8 = 0\]

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

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

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

\[1)\ 2^{x} = 4\]

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

\[x = 2.\]

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

\[x = 1.\]

\[Проверка:\]

\[\log_{5}\left( 4^{2} - 3 \cdot 2^{2} \right) =\]

\[= \log_{5}\left( 3 \cdot 2^{2} - 8 \right)\]

\[\log_{5}4 = \log_{5}4\]

\[x = 2 - корень.\]

\[\log_{5}(4 - 3 \cdot 2) = \log_{5}(3 \cdot 2 - 8)\]

\[\log_{5}( - 2) = \log_{5}( - 2)\]

\[a < 0;\]

\[x = 1 - не\ корень.\]

\[Ответ:x = 2.\]

\[\textbf{г)}\log_{4}\left( 9^{x} - 5 \cdot 3^{x} \right) =\]

\[= \log_{4}\left( 7 \cdot 3^{x} - 27 \right)\]

\[9^{x} - 5 \cdot 3^{x} = 7 \cdot 3^{x} - 27\]

\[\left( 3^{x} \right)^{2} - 12 \cdot 3^{x} + 27 = 0\]

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

\[t^{2} - 12t + 27 = 0\]

\[D_{1} = 36 - 27 = 9\]

\[t_{1} = 6 + 3 = 9;\]

\[t_{2} = 6 - 3 = 3.\]

\[1)\ 3^{x} = 9\]

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

\[x = 2.\]

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

\[x = 1.\]

\[Проверка:\]

\[\log_{4}\left( 9^{2} - 5 \cdot 3^{2} \right) =\]

\[= \log_{4}\left( 7 \cdot 3^{2} - 27 \right)\]

\[\log_{4}36 = \log_{4}36\]

\[x = 2 - корень.\]

\[\log_{4}(9 - 5 \cdot 3) =\]

\[= \log_{4}(7 \cdot 3 - 27)\]

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

\[x < 0;\]

\[x = 1 - не\ корень.\]

\[Ответ:x = 2.\]