A.S. level question on logarithms

Algebra Level 3

By forming a quadratic equation, show that there is only one value of x x which satisfies the equation log ⁡ 2 ( x + 7 ) − log ⁡ 2 ( x + 5 ) = 3 \log_2(x+7)-\log_2(x+5)=3

Find the answer x x .

Question from the AQA 2013 A level core 2 past paper question 8b).


The answer is -3.

Connor Buckley by Connor Buckley , 24, United Kingdom

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1 solution

Aditya Raut
Aug 25, 2014

Really awful formatting to understand, but I hope that it's intended to be

∙ 2 log ⁡ 2 ( x + 7 ) − log ⁡ 2 ( x + 5 ) = 3 ∴ log ⁡ 2 ( ( x + 7 ) 2 x + 5 ) = 3 ∴ ( x + 7 ) 2 ( x + 5 ) = 2 3 ∴ ( x + 7 ) 2 = 8 ( x + 5 ) ∴ x 2 + 14 x + 49 = 8 x + 40 ∴ x 2 + 6 x + 9 = 0 ∴ ( x + 3 ) 2 = 0 ∴ x = − 3 \displaystyle \bullet \quad 2\log_2 (x+7) - \log_2 (x+5) =3 \\ \therefore \log_2 \biggl( \dfrac{(x+7)^2}{x+5} \biggr) = 3\\ \therefore \dfrac{(x+7)^2}{(x+5)}= 2^3 \\ \therefore (x+7)^2=8(x+5) \\ \therefore x^2 +14x+49=8x+40 \\ \therefore x^2 +6x+9=0 \\ \therefore (x+3)^2=0 \\ \therefore x = \boxed{-3}

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