Решебник по алгебре 11 класс Никольский Параграф 7. Равносильность уравнений и неравенств Задание 30

\[\boxed{\mathbf{30.}}\]

\[\textbf{а)}\ 5^{x^{2} - 2x} > 5^{\log_{5}2^{x - 2}}\]

\[x^{2} - 2x > \log_{5}2^{x - 2}\]

\[x^{2} - 2x > (x - 2)\log_{5}2\]

\[x(x - 2) - (x - 2)\log_{5}2 > 0\]

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

\[x = \log_{5}2;\ \ x = 2;\]

\[\log_{5}2 < 2\log_{5}25\]

\[x < \log_{5}2;\]

\[x > 2.\]

\[\textbf{б)}\ 3^{x^{2} - x} < 3^{\log_{3}2^{1 - x}}\]

\[x^{2} - x < \log_{3}2^{1 - x}\]

\[x^{2} - x < (1 - x)\log_{3}2\]

\[x(x - 1) - (1 - x)\log_{3}2 < 0\]

\[x(x - 1) + (x - 1)\log_{3}2 < 0\]

\[(x - 1)\left( x + \log_{3}2 \right) < 0\]

\[x = 1;\]

\[x = - \log_{3}2;\]

\[- \log_{3}2 < x < 1.\]

\[\textbf{в)}\ 3^{x^{2} - x} > 3^{\log_{3}5^{x - 1}}\]

\[x^{2} - x > \log_{3}5^{x - 1}\]

\[x^{2} - x > (x - 1)\log_{3}5\]

\[x(x - 1) - (x - 1)\log_{3}5 > 0\]

\[(x - 1)\left( x - \log_{3}5 \right) > 0\]

\[x = 1;\ \ x = \log_{3}5;\]

\[x < 1;\]

\[x > \log_{3}5.\]

\[\textbf{г)}\ 7^{x^{2} - 5x} < 7^{\log_{7}6^{5 - x}}\]

\[x^{2} - 5x < \log_{7}6^{5 - x}\]

\[x^{2} - 5x < (5 - x)\log_{7}6\]

\[x(x - 5) - (5 - x)\log_{7}6 < 0\]

\[x(x - 5) + (x - 5)\log_{7}6 < 0\]

\[(x - 5)\left( x + \log_{7}6 \right) < 0\]

\[x = 5;\ \ x = - \log_{7}6;\]

\[- \log_{7}6 < x < 5.\]