<p>If \(2^a = 3\) and \(9^b = 4\), then the value of \(ab\) is</p>
Step-by-Step Solution
Key Concept: Write a = log_2 3 and b = log_9 4. a = log_2 3. Also b = log_9 4 = log_(3^2)(2^2) = log_3 2. Hence ab = (log_2 3)(log_3 2) = 1.
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. Write a = log_2 3 and b = log_9 4. a = log_2 3. Also b = log_9 4 = log_(3^2)(2^2) = log_3 2. Hence ab = (log_2 3)(log_3 2) = 1. Trap: Reduce log_9 4 carefully; both numerator and denominator squares cancel. 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: A