Решебник по алгебре и начала математического анализа 11 класс Алимов Задание 85

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\[1)\ 7^{x\sqrt{3}} = \sqrt{7}\]

\[7^{x\sqrt{3}} = 7^{\frac{1}{2}}\]

\[x\sqrt{3} = \frac{1}{2}\]

\[x = \frac{1}{2\sqrt{3}} = \frac{\sqrt{3}}{2 \bullet 3}\]

\[x = \frac{\sqrt{3}}{6}\]

\[Ответ:\ \ x = \frac{\sqrt{3}}{6}.\]

\[2)\ 25^{x\sqrt{2}} = 5\sqrt{5}\]

\[\left( 5^{2} \right)^{x\sqrt{2}} = 5^{1} \bullet 5^{\frac{1}{2}}\]

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

\[2x\sqrt{2} = \frac{3}{2}\]

\[x = \frac{3}{2 \bullet 2\sqrt{2}} = \frac{3\sqrt{2}}{2 \bullet 2 \bullet 2}\]

\[x = \frac{3\sqrt{2}}{8}\]

\[Ответ:\ \ x = \frac{3\sqrt{2}}{8}.\]

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

\[\left( 2^{\frac{1}{2}} \right)^{x} = 2^{1} \bullet 2^{\frac{1}{2}}\]

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

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

\[x = \frac{3}{2} \bullet 2\]

\[x = 3\]

\[Ответ:\ \ x = 3.\]

\[4)\ \left( \sqrt{3} \right)^{3x} = 3\sqrt{3}\]

\[\left( 3^{\frac{1}{2}} \right)^{3x} = 3^{1} \bullet 3^{\frac{1}{2}}\]

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

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

\[x = \frac{3}{2} \bullet \frac{2}{3}\]

\[x = 1\]

\[Ответ:\ \ x = 1.\]