Question

8．当函数$y={log_a}（{x^2}-a）$为减函数时，下列四个结论：
①$\left\{{\begin{array}{l}{0＜a＜1}\\{x＜-1}\end{array}}\right.$；②$\left\{{\begin{array}{l}{0＜a＜1}\\{x＞1}\end{array}}\right.$；③$\left\{{\begin{array}{l}{a＞1}\\{x＜-1}\end{array}}\right.$；④$\left\{{\begin{array}{l}{a＞1}\\{x＞1}\end{array}}\right.$
可以成立的是②．

Answer 1

分析 由题意可得$\left\{\begin{array}{l}{0＜a＜1}\\{x＞\sqrt{a}}\end{array}\right.$，或 $\left\{\begin{array}{l}{a＞1}\\{x＜-\sqrt{a}}\end{array}\right.$，结合所给的选项，可得结论．

解答 解：由函数$y={log_a}（{x^2}-a）$为减函数，可得$\left\{\begin{array}{l}{0＜a＜1}\\{x＞\sqrt{a}}\end{array}\right.$，或 $\left\{\begin{array}{l}{a＞1}\\{x＜-\sqrt{a}}\end{array}\right.$，
结合所给的选项，只有②满足，
故答案为：②．

点评 本题主要考查对数函数的单调性和特殊点，体现了等价转化的数学思想，属于基础题．