What is solution in inequality log base 2 (x+7)-2log base 2 x < 3?

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justaguide | College Teacher | (Level 2) Distinguished Educator

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The inequality `log_2(x+7)-2*log_2 x < 3` has to be solved.

`log_2(x+7)-2*log_2 x < 3`

=> `log_2(x+7)-log_2x^2 < 3`

=> `log_2((x+7)/x^2) < 3`

=> `(x+7)/x^2 < 2^3`

=> `(x+7)<8*x^2`

=> 8x^2 - x - 7 > 0

=> 8x^2 - 8x + 7x - 7 > 0

=> 8x(x - 1) + 7(x - 1) > 0

=> (8x + 7)(x - 1) > 0

This is true when

  • 8x + 7 > 0 and x - 1 > 0

=> x > -7/8 and x > 1

=> x > 1

  • 8x + 7 < 0 and x - 1 < 0

=> x < -7/8 and x < 1

=> x < -7/8

But the logarithm of a negative number is not defined. The second set of values can be eliminated.

The solution of the inequality is `(1, oo)` 

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