Решение уравнений (Вариант 1)

1. $$2^{3x+2} = 8$$ $$2^{3x+2} = 2^3$$ $$3x+2 = 3$$ $$3x = 1$$ $$x = \frac{1}{3}$$

   Ответ: $$x = \frac{1}{3}$$

2. $$3^{x-6} = \frac{1}{9}$$ $$3^{x-6} = 3^{-2}$$ $$x-6 = -2$$ $$x = 4$$

   Ответ: $$x = 4$$

3. $$5^{-x-2} = 125$$ $$5^{-x-2} = 5^3$$ $$-x-2 = 3$$ $$-x = 5$$ $$x = -5$$

   Ответ: $$x = -5$$

4. $$\left(\frac{1}{2}\right)^{4x-7} = 16$$ $$\left(2^{-1}\right)^{4x-7} = 2^4$$ $$2^{-4x+7} = 2^4$$ $$-4x+7 = 4$$ $$-4x = -3$$ $$x = \frac{3}{4}$$

   Ответ: $$x = \frac{3}{4}$$

5. $$81^{5-x} = \frac{1}{3}$$ $$(3^4)^{5-x} = 3^{-1}$$ $$3^{20-4x} = 3^{-1}$$ $$20-4x = -1$$ $$-4x = -21$$ $$x = \frac{21}{4}$$

   Ответ: $$x = \frac{21}{4}$$

6. $$5 \cdot 25^x = 125$$ $$5 \cdot (5^2)^x = 5^3$$ $$5^{2x+1} = 5^3$$ $$2x+1 = 3$$ $$2x = 2$$ $$x = 1$$

   Ответ: $$x = 1$$

7. $$(0.5)^{x^2-3} = 4$$ $$\left(\frac{1}{2}\right)^{x^2-3} = 2^2$$ $$(2^{-1})^{x^2-3} = 2^2$$ $$2^{-x^2+3} = 2^2$$ $$-x^2+3 = 2$$ $$-x^2 = -1$$ $$x^2 = 1$$ $$x = \pm 1$$

   Ответ: $$x = \pm 1$$

8. $$\left(\frac{1}{4}\right)^{x-5} = 256^x$$ $$(4^{-1})^{x-5} = (4^4)^x$$ $$4^{-x+5} = 4^{4x}$$ $$-x+5 = 4x$$ $$5x = 5$$ $$x = 1$$

   Ответ: $$x = 1$$

9. $$2^{3+x} = 0.4 \cdot 5^{3+x}$$ $$2^{3+x} = \frac{2}{5} \cdot 5^{3+x}$$ $$2^{3+x} = 2 \cdot 5^{2+x}$$ $$\frac{2^{3+x}}{2} = 5^{2+x}$$ $$2^{2+x} = 5^{2+x}$$ $$\left(\frac{2}{5}\right)^{2+x} = 1$$ $$\left(\frac{2}{5}\right)^{2+x} = \left(\frac{2}{5}\right)^0$$ $$2+x = 0$$ $$x = -2$$

   Ответ: $$x = -2$$