Question

подробно решить уравнение с одз

Answer 1

$log_{3}(x + 5) + log_{x + 5}(9) = 3$

$log_{3}(x + 5) + \frac{ log_{3}(9) }{ log_{3}(x + 5) } = 3$

$log_{3}(x + 5) + \frac{ log_{3}(3 {}^{2} ) }{ log_{3}(x + 5) } = 3$

$log_{3}(x + 5) + \frac{ 2}{ log_{3}(x + 5) } = 3$

$\fbox{t = log_{3}(x + 5) }$

$t + \frac{2}{t} = 3$

$t + \frac{2}{t} - 3 = 0$

$\frac{t {}^{2} + 2 - 3t}{t} = 0$

$t {}^{2} + 2 - 3t = 0$

$t {}^{2} - 3t + 2 = 0$

$t {}^{2} - t - 2t + 2 = 0$

$t(t - 1) - 2(t - 1) = 0$

$(t - 1)(t - 2) = 0$

$t - 1 = 0 \\ t - 2 = 0$

$t = 1 \\ t = 2$

$\fbox{t = log_{3}(x + 5) }$

$log_{3}(x + 5) = 1 \\ log_{3}(x + 5) = 2$

$x = - 2 \\ x = 4$

$\fbox{Otvet:x_{1} = - 2;x_{2} = 4}$