<p>If \(x_1\) and \(x_2\) are the roots of the equation \(e^{3/2}\,x^{2\ln x} = x^4\), then the product \(x_1x_2\) is</p>
Step-by-Step Solution
Key Concept: Write x^(2 ln x) as e^(2(ln x)^2) and take natural logs. Let t = ln x. Then 3/2 + 2t^2 = 4t, so 4t^2 - 8t + 3 = 0. Hence t1 + t2 = 2, and x1 x2 = e^(t1 + t2) = e^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. Write x^(2 ln x) as e^(2(ln x)^2) and take natural logs. Let t = ln x. Then 3/2 + 2t^2 = 4t, so 4t^2 - 8t + 3 = 0. Hence t1 + t2 = 2, and x1 x2 = e^(t1 + t2) = e^2. Trap: Convert everything to natural logarithms first; multiplying powers directly is awkward. 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