已知x,y为正数,log 2 (x+y+3)=log 2 (xy),求x+y的取值范围

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已知x,y为正数,log2(x+y+3)=log2(xy),求x+y的取值范围已知x,y为正数,log2(x+y+3)=log2(xy),求x+y的取值范围已知x,y为正数,log2(x+y+3)=l

已知x,y为正数,log 2 (x+y+3)=log 2 (xy),求x+y的取值范围
已知x,y为正数,log 2 (x+y+3)=log 2 (xy),求x+y的取值范围

已知x,y为正数,log 2 (x+y+3)=log 2 (xy),求x+y的取值范围
log 2 (x+y+3)=log 2 (xy),
x+y+3=xy
x+y+3=xy=0
(a-6)(a+2)>=0
a=6
x+y+3>0
x+y>-3
x>0,y>0
x+y>0
所以
取值范围是:
[6,正无穷)

根据log 2 (x+y+3)=log 2 (xy),既有:
x+y+3=xy,
又x+y>=2*根号(xy),所以:xy<=[(x+y)/2]^2
所以有x+y+3<=[(x+y)/2]^2,
得出(x+y-6)(x+y+2)>=0
所以x+y>=6或x+y<=-2,即属于(6,正无穷),(负无穷,-2)

x+y大于等于6