Question

若lg2,lg(2^x-1),lg(2^x+11)成等差数列,求x的值

Answer 1

由等差数列得到：lg2+lg(2^x+11)=2lg(2^x-1)

===> lg[2*(2^x+11)]=lg[(2^x-1)²]

===> 2*(2^x+11)=(2^x-1)²

令2^x=t（t＞0）,则：

2(t+11)=(t-1)²

===> 2t+22=t²-2t+1

===> t²-4t-21=0

===> (t+3)(t-7)=0

===> t=-3(＜0,舍去),t=7

即,2^x=7

所以,x=log7