Solve: `4(log)_(x/2)(sqrt(x))+2(log)_(4x)(x^2)=3(log)_(2x)(x^3)dot`

Explanation

`(4log_(2)sqrtx)/(log_(2)(x//2))+(2log_(2)(x^(2)))/(log_(2)(4x)) = (3log_(2)(x^(3)))/(log_(2)(2x))`
`(4 xxlog_(2)sqrtx)/(log_(2)(x//2))+(2log_(2)(x^(2)))/(log_(2)(x))=(9log_(2)(x))/(1+log_(2)(x))`
Let ` log_(2) x= t`. The given equation reduces to
`(2t)/(t-1)+(4t)/(t+2)=(9t)/(t+1)`
` rArr t=0 or 2/(t-1)+4/(t+2)=9/(t+1)`
`rArr (2t+4+4t-4)/((t-1)(t+2))=9/(t+1) or t = 0`
` rArr t^(2)+ t-6=0 or t=0`
` rArr (t+3)(t-2)=0 or t= 0`
` rArr t=0, 2, or-3`
`rArr x = 1, 4, 1//8`