III) Показательные уравнения Задание 4. Найдите корень уравнения: 1) 3x+8 = 3; 2) 9x-6 = 81; 3) 2-7+x = 16; 4) 5-1-x = 25; 5) 29-4x = 82x; 6) 83+2x = 642x

Задание 4.1

\(3^{x+8} = 3\)

\(3^{x+8} = 3^1\)

\(x+8 = 1\)

\(x = 1 - 8\)

\(x = -7\)

Ответ: -7

Задание 4.2

\(9^{x-6} = 81\)

\((3^2)^{x-6} = 3^4\)

\(3^{2(x-6)} = 3^4\)

\(2(x-6) = 4\)

\(2x - 12 = 4\)

\(2x = 16\)

\(x = 8\)

Ответ: 8

Задание 4.3

\(2^{-7+x} = 16\)

\(2^{-7+x} = 2^4\)

\(-7+x = 4\)

\(x = 4 + 7\)

\(x = 11\)

Ответ: 11

Задание 4.4

\(5^{-1-x} = 25\)

\(5^{-1-x} = 5^2\)

\(-1-x = 2\)

\(-x = 3\)

\(x = -3\)

Ответ: -3

Задание 4.5

\(2^{9-4x} = 8^{2x}\)

\(2^{9-4x} = (2^3)^{2x}\)

\(2^{9-4x} = 2^{6x}\)

\(9-4x = 6x\)

\(9 = 10x\)

\(x = \frac{9}{10}\)

\(x = 0.9\)

Ответ: 0.9

Задание 4.6

\(8^{3+2x} = 64^{2x}\)

\((2^3)^{3+2x} = (2^6)^{2x}\)

\(2^{3(3+2x)} = 2^{12x}\)

\(3(3+2x) = 12x\)

\(9+6x = 12x\)

\(9 = 6x\)

\(x = \frac{9}{6}\)

\(x = \frac{3}{2}\)

\(x = 1.5\)

Ответ: 1.5