Logarithm inequality

Question

log₂(x-3)+log(2x+6)<=2, (<=, less or equal)

Answer 1

log₂(x-3) + log₂(2x+6) ≤ 2

 

Note:  Since we can only find logarithms of positive numbers, we must have x > 3.

 

log₂[(x-3)(2x+6)] ≤ 2

 

log₂(2x²-18) ≤ 2

 

log₂[2(x²-9)] ≤ 2

 

log₂2 + log₂(x²-9) ≤ 2

 

1 + log₂(x²-9) ≤ 2

 

      log₂(x²-9) ≤ 1

 

      log₂(x²-9) ≤ log₂2

 

     x²-9 ≤ 2

 

     x² - 11 ≤ 0

 

(x - √11)(x + √11) ≤ 0

 

From the note above, we know that x > 3.

 

if 3 < x ≤ √11, (x-√11)(x+√11) ≤ 0

 

if x > √11, (x - √11)(x + √11) > 0

 

Solution set:  (x l 3 < x ≤ √11}