Solve `x^(log_(4) x)=2^(3(log_(4)x+3)`.

Explanation

To solve the equation \( x^{\log_{4} x} = 2^{3(\log_{4} x + 3)} \), we will follow these steps:

### Step 1: Rewrite the equation
We start with the original equation:
\[ x^{\log_{4} x} = 2^{3(\log_{4} x + 3)} \]

### Step 2: Simplify the right side
We can simplify the right side:
\[ 2^{3(\log_{4} x + 3)} = 2^{3 \log_{4} x + 9} = 2^{3 \log_{4} x} \cdot 2^9 \]
Since \(2^9 = 512\), we rewrite it as:
\[ x^{\log_{4} x} = 512 \cdot 2^{3 \log_{4} x} \]

### Step 3: Change the base of logarithm
Using the change of base formula, we can express \(\log_{4} x\) in terms of \(\log_{2} x\):
\[ \log_{4} x = \frac{\log_{2} x}{\log_{2} 4} = \frac{\log_{2} x}{2} \]
Thus, we can rewrite the equation:
\[ x^{\frac{\log_{2} x}{2}} = 512 \cdot 2^{3 \cdot \frac{\log_{2} x}{2}} \]

### Step 4: Simplify further
Now, substituting back into the equation:
\[ x^{\frac{\log_{2} x}{2}} = 512 \cdot 2^{\frac{3 \log_{2} x}{2}} = 512 \cdot (x^{\frac{3}{2}}) \]
This gives us:
\[ x^{\frac{\log_{2} x}{2}} = 512 \cdot x^{\frac{3}{2}} \]

### Step 5: Divide both sides by \(x^{\frac{3}{2}}\)
Assuming \(x \neq 0\), we can divide both sides by \(x^{\frac{3}{2}}\):
\[ x^{\frac{\log_{2} x}{2} - \frac{3}{2}} = 512 \]

### Step 6: Express 512 as a power of 2
We know that \(512 = 2^9\), so we can rewrite the equation:
\[ x^{\frac{\log_{2} x}{2} - \frac{3}{2}} = 2^9 \]

### Step 7: Take logarithm on both sides
Taking logarithm base 2 on both sides:
\[ \left(\frac{\log_{2} x}{2} - \frac{3}{2}\right) \log_{2} x = 9 \]

### Step 8: Let \(t = \log_{2} x\)
Let \(t = \log_{2} x\), then we have:
\[ \left(\frac{t}{2} - \frac{3}{2}\right) t = 9 \]
This simplifies to:
\[ \frac{t^2}{2} - \frac{3t}{2} - 9 = 0 \]

### Step 9: Multiply through by 2
Multiplying through by 2 to eliminate the fraction:
\[ t^2 - 3t - 18 = 0 \]

### Step 10: Factor the quadratic
Factoring the quadratic:
\[ (t - 6)(t + 3) = 0 \]
Thus, \(t = 6\) or \(t = -3\).

### Step 11: Solve for \(x\)
1. If \(t = 6\):
\[ \log_{2} x = 6 \implies x = 2^6 = 64 \]
2. If \(t = -3\):
\[ \log_{2} x = -3 \implies x = 2^{-3} = \frac{1}{8} \]

### Final Answer
The solutions are:
\[ x = 64 \quad \text{and} \quad x = \frac{1}{8} \]

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