Question

高中数学若方程log√2 x+log0.5(x-1)=log2 a有解,（根号2、0.5、2都是底）则实数a的取值范围是

Answer 1

x>1

log√2 x+log0.5(x-1)=2log2 x-log2 (x-1)=log2 [x^2/(x-1)]=log2 a

x^2/(x-1)=a

x^2-ax+a=0

Δ=a^2-4a=a(a-4)≥0

a≤0或a≥4

x1+x2=a>2

(x1-1)(x2-1)=x1x2-(x1+x2)+1=a-a+1=1>0

综上a≥4