If `x^((log_2 x)^(2)-6 log_2 x+11)=64` then x is equal to

Options

A. 2
B. 4
C. 6
D. 8

Answer

To solve the equation \( x^{(\log_2 x)^2 - 6 \log_2 x + 11} = 64 \), we will follow these steps:

### Step 1: Rewrite the equation
We know that \( 64 = 2^6 \). Therefore, we can rewrite the equation as:
\[ x^{(\log_2 x)^2 - 6 \log_2 x + 11} = 2^6 \]

### Step 2: Take logarithm base 2 on both sides
Taking logarithm base 2 on both sides gives us:
\[ (\log_2 x) \left((\log_2 x)^2 - 6 \log_2 x + 11\right) = 6 \]

### Step 3: Let \( p = \log_2 x \)
Substituting \( p \) for \( \log_2 x \), we have:
\[ p \left(p^2 - 6p + 11\right) = 6 \]
This simplifies to:
\[ p^3 - 6p^2 + 11p - 6 = 0 \]

### Step 4: Factor the cubic equation
We need to find the roots of the cubic equation \( p^3 - 6p^2 + 11p - 6 = 0 \). By testing possible rational roots, we find that \( p = 1 \) is a root.

### Step 5: Polynomial long division
Now we can factor out \( (p - 1) \) from the cubic polynomial:
\[ p^3 - 6p^2 + 11p - 6 = (p - 1)(p^2 - 5p + 6) \]

### Step 6: Factor the quadratic
Next, we factor the quadratic \( p^2 - 5p + 6 \):
\[ p^2 - 5p + 6 = (p - 2)(p - 3) \]

### Step 7: Find all roots
Thus, we have:
\[ (p - 1)(p - 2)(p - 3) = 0 \]
This gives us the roots:
\[ p = 1, \quad p = 2, \quad p = 3 \]

### Step 8: Convert back to \( x \)
Now we convert back to \( x \) using \( p = \log_2 x \):
1. For \( p = 1 \):
\[ \log_2 x = 1 \implies x = 2^1 = 2 \]
2. For \( p = 2 \):
\[ \log_2 x = 2 \implies x = 2^2 = 4 \]
3. For \( p = 3 \):
\[ \log_2 x = 3 \implies x = 2^3 = 8 \]

### Final Answer
The possible values of \( x \) are \( 2, 4, \) and \( 8 \).