251 1) 2^x + 2^(x-3) = 18; 3) 2⋅3^(x+1) - 6⋅3^(x-1) - 3^x = 9; 2) 3^x + 4⋅3^(x+1) = 13; 4) 5^(x+1) + 3⋅5^(x-1) - 6⋅5^x + 10 = 0.

1) $$2^x + 2^{x-3} = 18$$ $$2^x + \frac{2^x}{2^3} = 18$$ $$2^x + \frac{2^x}{8} = 18$$ $$2^x \cdot (1 + \frac{1}{8}) = 18$$ $$2^x \cdot \frac{9}{8} = 18$$ $$2^x = 18 \cdot \frac{8}{9}$$ $$2^x = 2 \cdot 8$$ $$2^x = 16$$ $$2^x = 2^4$$ $$x = 4$$ 2) $$3^x + 4 \cdot 3^{x+1} = 13$$ $$3^x + 4 \cdot 3^x \cdot 3 = 13$$ $$3^x + 12 \cdot 3^x = 13$$ $$13 \cdot 3^x = 13$$ $$3^x = 1$$ $$3^x = 3^0$$ $$x = 0$$ 3) $$2 \cdot 3^{x+1} - 6 \cdot 3^{x-1} - 3^x = 9$$ $$2 \cdot 3^x \cdot 3 - 6 \cdot \frac{3^x}{3} - 3^x = 9$$ $$6 \cdot 3^x - 2 \cdot 3^x - 3^x = 9$$ $$3^x (6 - 2 - 1) = 9$$ $$3^x \cdot 3 = 9$$ $$3^x = 3$$ $$3^x = 3^1$$ $$x = 1$$ 4) $$5^{x+1} + 3 \cdot 5^{x-1} - 6 \cdot 5^x + 10 = 0$$ $$5^x \cdot 5 + 3 \cdot \frac{5^x}{5} - 6 \cdot 5^x + 10 = 0$$ $$5^x (5 + \frac{3}{5} - 6) + 10 = 0$$ $$5^x (\frac{25 + 3 - 30}{5}) + 10 = 0$$ $$5^x (\frac{-2}{5}) + 10 = 0$$ $$5^x (\frac{-2}{5}) = -10$$ $$5^x = -10 : (\frac{-2}{5})$$ $$5^x = -10 \cdot (\frac{-5}{2})$$ $$5^x = 25$$ $$5^x = 5^2$$ $$x = 2$$ Ответ: 1) x = 4; 2) x = 0; 3) x = 1; 4) x = 2