The sum of integral values of $x$ which satisfy the equation $| x - 1 |^{(l o g x)^{2} - l o g x^{2}} = | x - 1 |^{3}$ is

2

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1002

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1001

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None of these

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Solution

The correct option is C $1001$
$| x - 1 |^{(log x)^{2} - log x^{2}} = | x - 1 |^{3}$
For, $x = 1$ , the equation satisfies,
Also,
$\Rightarrow (log x)^{2} - 2 log x = 3$
$\Rightarrow (log x)^{2} - 2 log x - 3 = 0$
$\Rightarrow (log x - 3) (log x + 1) = 0$
$\Rightarrow log x = - 1$
$\Rightarrow x = 10^{- 1}$
And $\Rightarrow log x = 3$
$\Rightarrow x = 10^{3}$
$\Rightarrow x = 1000$
Therefore, sum of integral value of $x$ is $1000 + 1 = 1001$

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