2log4{2log2[1+log2(1+log2x)]}=1\2

Question

2log4 {2log2[1+log2(1+log2 x)]}=1\2

Answer 1

方程式两边同乘以2,
4log4 {2log2[1+log2(1+log2 x)]}=1
方程式右边
1=log2 2……①
方程式左边
4log4 {2log2[1+log2(1+log2 x)]}
=log2^2{2log2[1+log2(1+log2 x)]}^4
=log2{2log2[1+log2(1+log2 x)]}^2……②
有①②得
{2log2[1+log2(1+log2 x)]}^2=2=(√2)^2
2log2[1+log2(1+log2 x)]=√2
[1+log2(1+log2 x)]^2=2^√2
1+log2(1+log2 x)=(√2)^√2
log2(1+log2 x)=(√2)^√2 -1
1+log2 x=2^[(√2)^√2 -1]
log2 x=2^[(√2)^√2 -1] -1
x=2^{2^[(√2)^√2 -1] -1}.