
The solution of the equation log2 (x−4) + log2 (x+4) = 3 . Express irrational solutions in exact form.

Answer to Problem 19CT
Solution:
The solution of log2 (x−4) + log2 (x+4) = 3 is x=2√6 .
Explanation of Solution
Given information:
The equation is log2 (x−4) + log2 (x+4) = 3 .
Consider the equation log2 (x−4) + log2 (x+4) = 3 .
The domain of the variable requires that x−4 > 0 and x+4 > 0 , so x > 4 and x > −4 .
This means, any solution must satisfy x > −4 .
To obtain exact solution, first express left side as a single logarithm.
By using logaM + logaN = loga(MN) ,
⇒log2 [(x−4)(x+4)] = 3
By using y=logax if and only if ay=x a > 0 a ≠ 1 ,
⇒(x−4)(x+4)=23
⇒x2−16=8
⇒x2−24=0
⇒x2=24
Taking square root of both sides,
⇒x=±√24
⇒x=±2√6
⇒x=2√6 or x=−2√6
Since x=−2√6 does not satisfy x > −4 ,
Hence, x=2√6 is the only solution of log2 (x−4) + log2 (x+4) = 3 .
Therefore, the solution of log2 (x−4) + log2 (x+4) = 3 is x=2√6 .
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