<p>The sum of the roots of the equation \(x + 1 - 2\log_2(3 + 2^x) + 2\log_4(10 - 2^{-x}) = 0\) is</p>
Step-by-Step Solution
Key Concept: Put t = 2^x and use 2 log_4 A = log_2 A. The equation becomes x + 1 = log_2[(3 + t)/(10 - 1/t)]. Rewriting x = log_2 t and simplifying gives a quadratic in t whose roots multiply to 11. Therefore the sum of x-roots is...
Notice that the cleanest route is to simplify the structure before computing. A clever move here is to translate the logarithmic statement into a friendlier algebraic form. Put t = 2^x and use 2 log_4 A = log_2 A. The equation becomes x + 1 = log_2[(3 + t)/(10 - 1/t)]. Rewriting x = log_2 t and simplifying gives a quadratic in t whose roots multiply to 11. Therefore the sum of x-roots is log_2 11. Trap: Convert all logarithms to the same base before substituting t = 2^x. Now, we invoke the power of the relevant logarithmic identity, simplify carefully, and finally verify the domain so that no extraneous answer survives.
Correct Answer: B