Страница 80 номер 219, ГДЗ по алгебре за 10-11 класс Алимов - ответы и решение
Решить уравнение (208—223). 1) \(7^{x - 2} = 3^{2 - x}\); 2) \(2^{x - 3} = 3^{3 - x}\); 3) \(3^{\frac{x + 2}{4}} = 5^{x + 2}\); 4) \(4^{\frac{x - 3}{2}} = 3^{2(x - 3)}\).
1) \(7^{x-2} = 3^{2-x}\)
\(7^{x-2} = \left(\dfrac{1}{3}\right)^{x-2},\)
\((7 \cdot 3)^{x-2} = 1,\)
\(21^{x-2} = 21^{0},\)
\(x - 2 = 0,\)
\(x = 2;\)
2) \(2^{x-3} = 3^{3-x}\)
\(2^{x-3} = \left(\dfrac{1}{3}\right)^{x-3},\)
\((2 \cdot 3)^{x-3} = 1,\)
\(6^{x-3} = 6^{0},\)
\(x - 3 = 0,\)
\(x = 3;\)
3) \(3^{\frac{x+2}{4}} = 5^{x+2}\)
\(\left(\sqrt[4]{3}\right)^{x+2} = 5^{x+2},\)
\(\left(\dfrac{\sqrt[4]{3}}{5}\right)^{x+2} = \left(\dfrac{\sqrt[4]{3}}{5}\right)^{0},\)
\(x + 2 = 0,\)
\(x = -2;\)
4) \(4^{\frac{x-3}{2}} = 3^{2(x-3)}\)
\(\left(\sqrt{4}\right)^{x-3} = 9^{x-3},\)
\(\left(\dfrac{\sqrt{4}}{9}\right)^{x-3} = \left(\dfrac{\sqrt{4}}{9}\right)^{0},\)
\(x - 3 = 0,\)
\(x = 3.\)
