Question

Log3(x3-x2-x+1) -log3(x-1) -log3(x+1) =2 s

Answer 1

SOLUTION

TO DETERMINE

To solve for x

$\displaystyle \sf{ log_{3}( {x}^{3} - {x}^{2} - x + 1 ) - log_{3}( x - 1 ) - log_{3}( x + 1 ) = 2}$

EVALUATION

$\displaystyle \sf{ log_{3}( {x}^{3} - {x}^{2} - x + 1 ) - log_{3}( x - 1 ) - log_{3}( x + 1 ) = 2}$

$\displaystyle \sf{ \implies log_{3}[ {x}^{2}(x - 1) - (x - 1 )] -[ log_{3}( x - 1 ) + log_{3}( x + 1 )] = 2}$

$\displaystyle \sf{ \implies log_{3}[({x}^{2} - 1)(x - 1)] - log_{3}( x - 1 )(x + 1) = 2}$

$\displaystyle \sf{ \implies log_{3} \bigg[ \frac{({x}^{2} - 1)(x - 1)}{( x - 1 )(x + 1)} \bigg] = 2}$

$\displaystyle \sf{ \implies log_{3} \bigg[ \frac{(x + 1)(x - 1)(x - 1)}{( x - 1 )(x + 1)} \bigg] = 2}$

$\displaystyle \sf{ \implies log_{3} \bigg[ (x - 1) \bigg] = 2}$

$\displaystyle \sf{ \implies (x - 1) = {3}^{2} }$

$\displaystyle \sf{ \implies (x - 1) = 9 }$

$\displaystyle \sf{ \implies x = 9 + 1 }$

$\displaystyle \sf{ \implies x = 10}$

Final Answer :

Hence the required solution is x = 10

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