Решите показательные уравнения: 1. 3 \(\cdot\) 27^{x-2} = 243 2. 2 \(\cdot\) 8^{3x-1} = 64 3. 5 \(\cdot\) 25^{x+1} = 625 4. 4 \(\cdot\) 16^{x-3} = 256 5. 3 \(\cdot\) 9^{2x+1} = 2187 6. 2^{x^2-3x} = 16 7. 5^{x^2-2x} = 125 8. 3^{x^2+2x} = 27 9. 2^{x^2-5x+6} = 1 10. 5^{x^2-7x+12} = 1 11. 3^{x+4} = 9^x 12. 2^{2x+3} = 8^x 13. 4^{x-1} = 2^{x+3} 14. 27^{x+2} = 9^{2x-1} 15. 5^{3x-1} = 25^{x+2}

Решение:

Для решения показательных уравнений приведём обе части уравнения к одному основанию.

1. \(3 \cdot 27^{x-2} = 243\)
   \(27^{x-2} = \frac{243}{3}\)
   \(3^{3(x-2)} = 3^5\)
   \(3x - 6 = 5\)
   \(3x = 11\)
   \(x = \frac{11}{3}\)
2. \(2 \cdot 8^{3x-1} = 64\)
   \(8^{3x-1} = \frac{64}{2}\)
   \(2^{3(3x-1)} = 2^5\)
   \(9x - 3 = 5\)
   \(9x = 8\)
   \(x = \frac{8}{9}\)
3. \(5 \cdot 25^{x+1} = 625\)
   \(25^{x+1} = \frac{625}{5}\)
   \(5^{2(x+1)} = 5^3\)
   \(2x + 2 = 3\)
   \(2x = 1\)
   \(x = \frac{1}{2}\)
4. \(4 \cdot 16^{x-3} = 256\)
   \(16^{x-3} = \frac{256}{4}\)
   \(4^{x-3} = 4^2\)
   \(x - 3 = 2\)
   \(x = 5\)
5. \(3 \cdot 9^{2x+1} = 2187\)
   \(9^{2x+1} = \frac{2187}{3}\)
   \(3^{2(2x+1)} = 3^7\)
   \(4x + 2 = 7\)
   \(4x = 5\)
   \(x = \frac{5}{4}\)
6. \(2^{x^2-3x} = 16\)
   \(2^{x^2-3x} = 2^4\)
   \(x^2 - 3x = 4\)
   \(x^2 - 3x - 4 = 0\)
   \(x_1 = 4, x_2 = -1\)
7. \(5^{x^2-2x} = 125\)
   \(5^{x^2-2x} = 5^3\)
   \(x^2 - 2x = 3\)
   \(x^2 - 2x - 3 = 0\)
   \(x_1 = 3, x_2 = -1\)
8. \(3^{x^2+2x} = 27\)
   \(3^{x^2+2x} = 3^3\)
   \(x^2 + 2x = 3\)
   \(x^2 + 2x - 3 = 0\)
   \(x_1 = 1, x_2 = -3\)
9. \(2^{x^2-5x+6} = 1\)
   \(2^{x^2-5x+6} = 2^0\)
   \(x^2 - 5x + 6 = 0\)
   \(x_1 = 2, x_2 = 3\)
10. \(5^{x^2-7x+12} = 1\)
    \(5^{x^2-7x+12} = 5^0\)
    \(x^2 - 7x + 12 = 0\)
    \(x_1 = 3, x_2 = 4\)
11. \(3^{x+4} = 9^x\)
    \(3^{x+4} = 3^{2x}\)
    \(x + 4 = 2x\)
    \(x = 4\)
12. \(2^{2x+3} = 8^x\)
    \(2^{2x+3} = 2^{3x}\)
    \(2x + 3 = 3x\)
    \(x = 3\)
13. \(4^{x-1} = 2^{x+3}\)
    \(2^{2(x-1)} = 2^{x+3}\)
    \(2x - 2 = x + 3\)
    \(x = 5\)
14. \(27^{x+2} = 9^{2x-1}\)
    \(3^{3(x+2)} = 3^{2(2x-1)}\)
    \(3x + 6 = 4x - 2\)
    \(x = 8\)
15. \(5^{3x-1} = 25^{x+2}\)
    \(5^{3x-1} = 5^{2(x+2)}\)
    \(3x - 1 = 2x + 4\)
    \(x = 5\)

Ответ: 1. \(x = \frac{11}{3}\); 2. \(x = \frac{8}{9}\); 3. \(x = \frac{1}{2}\); 4. \(x = 5\); 5. \(x = \frac{5}{4}\); 6. \(x_1 = 4, x_2 = -1\); 7. \(x_1 = 3, x_2 = -1\); 8. \(x_1 = 1, x_2 = -3\); 9. \(x_1 = 2, x_2 = 3\); 10. \(x_1 = 3, x_2 = 4\); 11. \(x = 4\); 12. \(x = 3\); 13. \(x = 5\); 14. \(x = 8\); 15. \(x = 5\).