设正数x,y满足log2(x+y+3)=log2x+log2y,则x+y的取值范围是()A.(0,6]B.[6,+∞)C.[1+7,+∞)D.(0,1+7]-数学

题目简介

设正数x,y满足log2(x+y+3)=log2x+log2y,则x+y的取值范围是()A.(0,6]B.[6,+∞)C.[1+7,+∞)D.(0,1+7]-数学

题目详情

设正数x,y满足log2(x+y+3)=log2x+log2y,则x+y的取值范围是(  )
A.(0,6]B.[6,+∞)C.[1+
7
,+∞)
D.(0,1+
7
]
题型:单选题难度:中档来源:不详

答案

由正数x,y满足log2(x+y+3)=log2x+log2y,∴x+y+3=xy,
xy≤(class="stub"x+y
2
)2
,则x+y+3
(x+y)2
4
.当且仅当x=y>0时取等号.
令x+y=t,则t+3≤
t2
4
化为t2-4t-12≥0,解得t≥6或t≤-2.
∵t>0,∴取t≥6.
故选B.

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