Solve:` 27^(log_(3)root(3)(x^(2)-3x+1) )=(log_(2)(x-1))/(|log_(2)(x-1)|)`.

Answer

x = 3

Explanation

We have `27^(log_(3)root(3)(x^(2)-3x+1) )=(log_(2)(x-1))/(|log_(2) (x-1)|)`
We must have `x-1 gt 0 and x - 1 ne 1`
` :. X gt and x ne 2`
If ` 1 lt x lt 2, log_(2) (x-1) lt 0`
`:. (log_(2)(x-1))/(|log_(2)(x-1)|) =- 1`
This is not possible as L.H.S. ` gt 0` always.
If ` x gt 2," then "(log_(2)(x-1))/(|log_(2)(x-1)|) = 1`
`:." Equation reduces to "27^(log_(3)root(3)(x^(2)-3x+1))=1`
`:gt x^(2)- 3x+1 = 1`
` rArr x = 0, 3," but "x ne 0`
` :. x = 3` is the only solution.