\(2 \log_{3} 2 / \left(2 \log_{3} 2 - 1\right)\)
\(2 / \left(2 - \log_{2} 3\right)\)
\(1 / \left(1 - \log_{4} 3\right)\)
\(2 \log_{2} 3 / \left(2 \log_{2} 3 - 1\right)\)
Step-by-Step Solution
Key Concept: Take logs and isolate x, then rewrite the same value in different equivalent forms. From x log 3 = (x - 1)log 4, we get x = log 4 /(log 4 - log 3). Rewriting this using different bases shows that options A, B and C ar...
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. Take logs and isolate x, then rewrite the same value in different equivalent forms. From x log 3 = (x - 1)log 4, we get x = log 4 /(log 4 - log 3). Rewriting this using different bases shows that options A, B and C are equivalent to the same value, while D is not. Trap: All correct options are algebraically equivalent forms of the same x. 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, B, C