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
Let \( u = (\log_2 x)^2 - 6\log_2 x + 12 \) where \(x\) is a real number. Then the equation \(x^u = 256\), has
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For exponential equations, try logarithmic transformations to simplify the equation and find the solutions.
Updated On: Mar 20, 2026
- no solution for \(x\)
- exactly one solution for \(x\)
- exactly two distinct solutions for \(x\)
- exactly three distinct solutions for \(x\)
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The Correct Option is C
Solution and Explanation
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 \(x\).
\[
\boxed{\text{Two distinct solutions for } x}
\]
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