<p>If \(a^4 b^5 = 1\), then the value of \(\log_a(a^5 b^4)\) equals</p>
Step-by-Step Solution
Key Concept: Use a^4 b^5 = 1 to express b in terms of a. From b^5 = a^-4, we get b = a^(-4/5). Hence a^5 b^4 = a^(5 - 16/5) = a^(9/5), so the logarithm is 9/5.
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. Use a^4 b^5 = 1 to express b in terms of a. From b^5 = a^-4, we get b = a^(-4/5). Hence a^5 b^4 = a^(5 - 16/5) = a^(9/5), so the logarithm is 9/5. Trap: After converting b into a power of a, combine exponents before taking the logarithm. 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