<p>The sum of all solutions of the equation \(8^{2x} - 16\cdot 8^x + 48 = 0\) is</p>
Step-by-Step Solution
Key Concept: Let y = 8^x to get a quadratic. Then y^2 - 16y + 48 = 0 gives y = 4 or 12. Hence x = log_8 4 and log_8 12. Their sum is log_8(48) = 1 + log_8 6.
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. Let y = 8^x to get a quadratic. Then y^2 - 16y + 48 = 0 gives y = 4 or 12. Hence x = log_8 4 and log_8 12. Their sum is log_8(48) = 1 + log_8 6. Trap: When adding logarithms with the same base, combine them into the log of a product. 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: C