Решение показательных уравнений

Решение показательных уравнений. 1. \( 4^x = 64 \) \[ 4^x = 4^3 \] \[ x = 3 \] Ответ: 3. 2. \( 3^x = \frac{1}{9} \) \[ 3^x = 3^{-2} \] \[ x = -2 \] Ответ: -2. 3. \( 25^{-x} = \frac{1}{5} \) \[ (5^2)^{-x} = 5^{-1} \] \[ 5^{-2x} = 5^{-1} \] \[ -2x = -1 \] \[ x = 0,5 \] Ответ: 0,5. 4. \( (0,5)^x = \frac{1}{64} \) \[ (\frac{1}{2})^x = (\frac{1}{2})^6 \] \[ x = 6 \] Ответ: 6. 5. \( \sqrt[3]{128} = 4^{2x} \) \[ (2^7)^{\frac{1}{3}} = (2^2)^{2x} \] \[ 2^{\frac{7}{3}} = 2^{4x} \] \[ \frac{7}{3} = 4x \] \[ x = \frac{7}{12} \] Ответ: \( \frac{7}{12} \). 6. \( 16 \cdot 8^{2+3x} = 1 \) \[ 2^4 \cdot (2^3)^{2+3x} = 2^0 \] \[ 2^4 \cdot 2^{6+9x} = 2^0 \] \[ 2^{10+9x} = 2^0 \] \[ 10 + 9x = 0 \] \[ 9x = -10 \] \[ x = -1\frac{1}{9} \] Ответ: \( -1\frac{1}{9} \). 7. \( 128 \cdot 16^{2x+1} = 8^{3-2x} \) \[ 2^7 \cdot (2^4)^{2x+1} = (2^3)^{3-2x} \] \[ 2^7 \cdot 2^{8x+4} = 2^{9-6x} \] \[ 2^{8x+11} = 2^{9-6x} \] \[ 8x + 11 = 9 - 6x \] \[ 14x = -2 \] \[ x = -\frac{1}{7} \] Ответ: \( -\frac{1}{7} \). 8. \( 10^{x^2+x-2} = 1 \) \[ 10^{x^2+x-2} = 10^0 \] \[ x^2 + x - 2 = 0 \] По теореме Виета: \[ x_1 = -2, x_2 = 1 \] Ответ: -2; 1. 9. \( 3^{x^2-4x-0,5} = 81\sqrt{3} \) \[ 3^{x^2-4x-0,5} = 3^4 \cdot 3^{0,5} \] \[ 3^{x^2-4x-0,5} = 3^{4,5} \] \[ x^2 - 4x - 0,5 = 4,5 \] \[ x^2 - 4x - 5 = 0 \] По теореме Виета: \[ x_1 = 5, x_2 = -1 \] Ответ: 5; -1. 10. \( (\frac{2}{3})^x \cdot (\frac{9}{8})^x = \frac{27}{64} \) \[ (\frac{2}{3} \cdot \frac{9}{8})^x = \frac{27}{64} \] \[ (\frac{3}{4})^x = (\frac{3}{4})^3 \] \[ x = 3 \] Ответ: 3. 11. \( (0,4)^{x-1} = (6,25)^{6x-5} \) \[ (\frac{2}{5})^{x-1} = ((\frac{5}{2})^2)^{6x-5} \] \[ (\frac{2}{5})^{x-1} = (\frac{2}{5})^{-2(6x-5)} \] \[ x - 1 = -12x + 10 \] \[ 13x = 11 \] \[ x = \frac{11}{13} \] Ответ: \( \frac{11}{13} \). 12. \( 2^{x+4} - 2^x = 120 \) \[ 2^x(2^4 - 1) = 120 \] \[ 2^x(16 - 1) = 120 \] \[ 2^x \cdot 15 = 120 \] \[ 2^x = 8 \] \[ 2^x = 2^3 \] \[ x = 3 \] Ответ: 3. 13. \( 5^{2x+1} - 3 \cdot 5^{2x-1} = 550 \) \[ 5^{2x-1}(5^2 - 3) = 550 \) \[ 5^{2x-1}(25 - 3) = 550 \] \[ 5^{2x-1} \cdot 22 = 550 \] \[ 5^{2x-1} = 25 \] \[ 5^{2x-1} = 5^2 \] \[ 2x - 1 = 2 \] \[ 2x = 3 \] \[ x = 1,5 \] Ответ: 1,5. 14. \( \sqrt{3^x} \cdot 5^{\frac{x}{2}} = 225 \) \[ 3^{\frac{x}{2}} \cdot 5^{\frac{x}{2}} = 225 \] \[ (3 \cdot 5)^{\frac{x}{2}} = 15^2 \] \[ 15^{\frac{x}{2}} = 15^2 \] \[ \frac{x}{2} = 2 \] \[ x = 4 \] Ответ: 4. 15. \( 2^{x^3-9x} = 1 \) \[ 2^{x^3-9x} = 2^0 \] \[ x^3 - 9x = 0 \] \[ x(x^2 - 9) = 0 \] \[ x(x - 3)(x + 3) = 0 \] \[ x_1 = 0, x_2 = 3, x_3 = -3 \] Ответ: 0; 3; -3.