<p>Let \(a\) and \(b\) be real numbers greater than \(1\). If there exists a positive real number \(c \ne 1\) such that \(2(\log_a c + \log_b c) = 9\log_{ab} c\), then the possible value(s) of \(\log_a b\) is(are)</p>
Step-by-Step Solution
Key Concept: Put p = log_a c and q = log_b c and use log_(ab) c = pq/(p+q). After simplification, the condition reduces to a quadratic in k = log_a b. Solving gives k = 1/2 or 2. Thus the possible values are 1/2 and 2.
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 p = log_a c and q = log_b c and use log_(ab) c = pq/(p+q). After simplification, the condition reduces to a quadratic in k = log_a b. Solving gives k = 1/2 or 2. Thus the possible values are 1/2 and 2. Trap: Use reciprocal and product relations of logs before introducing k = log_a b. 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, D