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Find all real solutions to \(2 \log_2 (x + 5) = \log_2 (x - 9) + \log_2 (x + 53) + 1.\) Enter all the solutions, separated by commas.

 Apr 1, 2019
 #1
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\(2\log_2(x+5)-\log_2(x-9)-\log_2(x+53)=1\\ \log_2\left(\dfrac{(x+5)^2}{(x-9)(x+53)}\right)=1\\ \dfrac{(x+5)^2}{(x-9)(x+53)} = 2\)

 

\(x^2+10x+25 = 2x^2+88x-954\\ x^2+78x-979=0\\ (x+89)(x-11) = 0\\ x = -89, 11\)

 

\(\text{However }x=-89 \text{ ends up lying outside the domain of the log functions in the original equation}\\ \text{so it's not a valid solution. Thus }x=11 \text{ is the only solution}\)

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 Apr 1, 2019

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