<p>If \(\log_2(4 + \log_3 x) = 3\), then the sum of digits of \(x\) is</p>
Step-by-Step Solution
Key Concept: Undo the outer logarithm and then the inner logarithm. log_2(4 + log_3 x) = 3 gives 4 + log_3 x = 8, so log_3 x = 4. Hence x = 81 and the sum of digits is 8 + 1 = 9.
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. Undo the outer logarithm and then the inner logarithm. log_2(4 + log_3 x) = 3 gives 4 + log_3 x = 8, so log_3 x = 4. Hence x = 81 and the sum of digits is 8 + 1 = 9. Trap: The inner term is log_3 x, not x itself. 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