Question

Показникові рівняння і нерівності пожалуйста 30 баллов

Answer 1

Ответ:

1.

${25}^{x} + 25 = 26 \times {5}^{x} \\ \\ {5}^{x} = t \\ \\ t {}^{2} + 25 = 26t \\ t {}^{2} - 26 t + 25 = 0 \\ D = 169 - 25 = 144\\ t_1 = \frac{13 + 12}{1} = 25 \\ t_2 = 1 \\ \\ {5}^{x} = 25 \\ x_1 = 2 \\ \\ {5}^{x} = 1 \\ x_2 = 0$

Ответ: 0; 2.

2.

$5 \times {4}^{x} - 7 \times {10}^{x} + 2 \times {25}^{x} = 0 \\ 5 \times {2}^{2x} - 7 \times {2}^{x} \times {5}^{x} + 2 \times {5}^{2x} = 0 \\ | \div ( {2}^{x} \times {5}^{x} ) \\ 5 \times \frac{2 {}^{x} }{ {5}^{x} } - 7 + 2 \times \frac{ {5}^{x} }{ {2}^{x} } = 0 \\ \\ ( \frac{2}{5} ) {}^{x} = t \\ \\ 5t - 7 + \frac{2}{t} = 0 \\ \\ t\ne0 \\ \\ 5 {t}^{2} - 7 t + 2 = 0 \\ D = 49 - 40 = 9\\ t_1 = \frac{7 + 3}{10} = 1 \\ t_2 = \frac{4}{10} = \frac{2}{5} \\ \\ {( \frac{2}{5} })^{x} = 1 \\ x_1 = 0 \\ \\ ( \frac{2}{5} ) {}^{x} = \frac{2}{5} \\ x_2 = 1$

Ответ: 0; 1

3.

${3}^{x + 3} + 5 \times {3}^{x - 1} = 86 \\ {3}^{x - 1} ( {3}^{4} + 5) = 86 \\ {3}^{x - 1} \times 86 = 86 \\ {3}^{x - 1} = 1 \\ x - 1 = 0 \\ x = 1$

4.

${4}^{x + 1} + {4}^{x} \geqslant 320 \\ {4}^{x} (4 + 1) \geqslant 320 \\ {4}^{x} \times 5 \geqslant 320 \\ {4}^{x} \geqslant 64 \\ x \geqslant 3$