Question

Решите уравнения на изображении

Answer 1

Ответ:

$1)5x - 8 = {( \frac{1}{2}) }^{ - 1} \\ 5x - 8 = 2 \\ 5x = 10 \\ x = 2$

$2)2x + 3 = x + 5 \\ x = 2$

$3) log_{2}(x + 1) = log_{2}(2) + log_{2}(x) \\ log_{2}(x + 1) = log_{2}(2x) \\ x + 1 = 2x \\ - x = - 1 \\ x = 1$

$4) log_{ \frac{1}{5} }(x) = t \\ {t}^{2} + t - 6 =0 \\ d = 1 + 24 = 25 \\ t1 = ( - 1 + 5) \div 2 = 2 \\ t2 = - 3$

$log_{ \frac{1}{5} }(x) = 2 \\ x1 = \frac{1}{25} \\ log_{ \frac{1}{5} }(x) = - 3 \\ x2 = 125$

$5) log_{7}( \frac{12 - {x}^{2} }{2 - x} ) = log_{7}(2x + 9) \\ \frac{12 - {x}^{2} }{2 - x} = 2x + 9 \\ 12 - {x}^{2} = (2x + 9)(2 - x) \\ 12 - {x}^{2} = 4x - 2 {x}^{2} + 18 - 9x \\ {x}^{2} + 5x - 6 = 0 \\ d = 25 + 24 = 49 \\ x1 = ( - 5 + 7) \div 2 = 1 \\ x2 = - 6$

ОДЗ:

$12 - {x}^{2} > 0 \\ 2 - x > 0 \: \: \: \\ 2x + 9 > 0$

$(2 \sqrt{3} - x)(2 \sqrt{ 3} + x) > 0 \\ x < 2 \\ x > - 4.5$

Получаем: х принадлежит

$( - 2 \sqrt{3}; 2)$

корень -6 не подходит.

Ответ: 1.