Question

Let \( u = (\log_2 x)^2 - 6\log_2 x + 12 \) where \(x\) is a real number. Then the equation \(x^u = 256\), has

• A. no solution for \(x\)
• B. exactly one solution for \(x\)
• C. exactly two distinct solutions for \(x\)
• D. exactly three distinct solutions for \(x\)

Hint:

For exponential equations, try logarithmic transformations to simplify the equation and find the solutions.

Answer 1

The Correct Option is C

We start by solving the equation \(x^u = 256\). We can rewrite \(256 = 2^8\), so: \[ x^{(\log_2 x)^2 - 6\log_2 x + 12} = 2^8. \] Solving this equation yields two distinct solutions for \(x\). \[ \boxed{\text{Two distinct solutions for } x} \]