函数y=log2x+logx（2x）的值域是．

Question

函数y=log₂x+log_x（2x）的值域是______．

Answer 1

∵y=log₂x+log_x（2x）=log₂x+log_xx+log_x2
=log₂x+log_x2+1=log2x+(1)/(log2x)+1
令t=log₂x，∵x＞0且x≠1，∴t＞0或t＜0．
∴t+(1)/(t)≥2，或t+(1)/(t)≤−2
∴y=t+(1)/(t)+1≤-1，或y≥3，
故答案为：（-∞，-1]∪[3，+∞）．