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Question

If |x1|((log10x)2log10x2)=|x1|3, then 
  1. x=110 is a solution for the equation
  2. x=2 is a solution for the equation
  3. Number of values of x satisfying the equation is 4
  4. x=1 is a solution for the equation


Solution

The correct options are
A x=110 is a solution for the equation
B x=2 is a solution for the equation
|x1|((log10x)2log10x2)=|x1|3
For the log to be defined,
x>0
If am=a3
Then
(i) m=3(log10x)2log10x2=3
Assuming t=log10x
t22t3=0
(t3)(t+1)=0
log10x=1,3
x=101,103

(ii) a=1|x1|=1x=0,2x=2   (x>0)

(iii) a=0,m0
a=0|x1|=0x=1
Checking the value of m at x=1,
m=(log101)2log1012=0
Which contradict our assumption

Therefore the required values are
x=110,2,103

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