<p>If \((\log_\beta \alpha)^2 + (\log_\alpha \beta)^2 = 79\), then the value of \((\log_\beta \alpha) + (\log_\alpha \beta)\) can be</p>
Step-by-Step Solution
Key Concept: Let t = log_beta alpha so log_alpha beta = 1/t. Then t^2 + 1/t^2 = 79, so (t + 1/t)^2 = 81. Hence t + 1/t = \pm9. Therefore the possible values are -9 and 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. Let t = log_beta alpha so log_alpha beta = 1/t. Then t^2 + 1/t^2 = 79, so (t + 1/t)^2 = 81. Hence t + 1/t = \pm9. Therefore the possible values are -9 and 9. Trap: Use the reciprocal relation between log_beta alpha and log_alpha beta immediately. 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: B, C