Question

Решите неравенство
log_x^2(x−4)2 ≤ 0

Answer 1

Метод рационализации.

$log_{x^2}(x-4)^2\leq 0\\\\log_{x^2}(x-4)^2-log_{x^2}1\leq 0 \; \;\; \Leftrightarrow \; \; \left\{\begin{array}{l}x^2>0\; ,\; x^2\ne 1\\(x-4)^2>0\\((x-4)^2-1)(x^2-1)\leq 0\end{array}\right\\\\\\(x^2-8x+15)(x-1)(x+1)\leq 0\\\\(x-3)(x-5)(x-1)(x+1)\leq 0\\\\znaki:\; \;+++[-1\, ]---[\, 1\, ]+++[\, 3\, ]---[\, 5\, ]+++\\\\x\in [-1,1\, ]\cup [\, 3,5\, ]\\\\\left\{\begin{array}{ccc}x\ne 0\; ,\; x\ne \pm 1\\x\ne 4\\x\in [-1,1\, ]\cup [\, 3,5\, ]\end{array}\right\; \; \; \Rightarrow \; \; \underline {\; x\in (-1,1)\cup [\, 3,4)\cup (4,5\, ]\; }$