Решить уравнение log2 (x-1) - log2 (x^2-x-16) = 0

$log_{2}(x-1) - log_{2}(x^{2}-x -16) = 0$

$log_{2}(\frac{x-1}{x^{2}-x-16 }) =0$

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ОДЗ:

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x² - x - 16>0

x² - x - 16=0

D = (-1)²- 4 * (-16) = 1 + 64 = 65

$x_{1} = \frac{1+\sqrt{65} }{2}$ ≈ 4,5

$x_{2} = \frac{1-\sqrt{65} }{2}$ ≈ -3,5

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$\frac{x-1}{x^{2}-x-16 } =1$ | * ( x² - x - 16 )

x-1 = x² - x - 16

x² - 2x - 15 = 0

D = (-2)² - 4 * (-15) = 4 + 60 = 64 = 8²

$x_{1} = \frac{2+8}{2*1}= \frac{10}{2} = 5$

$x_{2} = \frac{2-8}{2*1}= -\frac{6}{2} = -3$ - посторонний корень

Ответ: x = 5