<p>The equation \(x^{(\log_3 x)^2 - \frac{9}{2}\log_3 x + 5} = 3\sqrt{3}\) has</p>
exactly three real solutions
\(at least one real solution\)
exactly one irrational solution
\(at least one imaginary root\)
Step-by-Step Solution
Key Concept: Put t = log_3 x so x = 3^t and compare exponents of 3. The equation becomes 3^[t(t^2 - 9t/2 + 5)] = 3^(3/2), so t(t^2 - 9t/2 + 5) = 3/2. This cubic factors into three real roots, giving three positive real values of x...
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 t = log_3 x so x = 3^t and compare exponents of 3. The equation becomes 3^[t(t^2 - 9t/2 + 5)] = 3^(3/2), so t(t^2 - 9t/2 + 5) = 3/2. This cubic factors into three real roots, giving three positive real values of x, among which exactly one is irrational. Hence A, B and C are true. Trap: Because x is inside a logarithm and also the base, x must stay positive throughout. 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