Solve: `x^(log_(4)3)+3^(log_(4)x)=18`.

Answer

To solve the equation \( x^{\log_{4}3} + 3^{\log_{4}x} = 18 \), we can follow these steps:

### Step 1: Rewrite the equation
Let \( y = x^{\log_{4}3} \). Then, the equation becomes:
\[ y + 3^{\log_{4}x} = 18 \]

### Step 2: Express \( 3^{\log_{4}x} \) in terms of \( y \)
Using the property of logarithms, we can express \( 3^{\log_{4}x} \) as follows:
\[ 3^{\log_{4}x} = 3^{\frac{\log_{3}x}{\log_{3}4}} = \left(3^{\log_{3}x}\right)^{\frac{1}{\log_{3}4}} = x^{\frac{1}{\log_{3}4}} \]
Thus, we can rewrite the equation:
\[ y + x^{\frac{1}{\log_{3}4}} = 18 \]

### Step 3: Substitute \( y \) back into the equation
Now substituting \( y = x^{\log_{4}3} \) back into the equation gives us:
\[ x^{\log_{4}3} + x^{\frac{1}{\log_{3}4}} = 18 \]

### Step 4: Simplify the equation
We know that \( \log_{4}3 = \frac{1}{\log_{3}4} \), so we can express \( x^{\frac{1}{\log_{3}4}} \) as \( x^{\log_{4}3} \). Thus, we can rewrite:
\[ x^{\log_{4}3} + x^{\log_{4}3} = 18 \]
This simplifies to:
\[ 2x^{\log_{4}3} = 18 \]

### Step 5: Solve for \( x^{\log_{4}3} \)
Dividing both sides by 2 gives:
\[ x^{\log_{4}3} = 9 \]

### Step 6: Take logarithm on both sides
Taking logarithm base 4 on both sides:
\[ \log_{4}(x^{\log_{4}3}) = \log_{4}(9) \]
Using the power rule of logarithms:
\[ \log_{4}3 \cdot \log_{4}x = \log_{4}(9) \]

### Step 7: Solve for \( \log_{4}x \)
Now, we can express \( \log_{4}x \) as:
\[ \log_{4}x = \frac{\log_{4}(9)}{\log_{4}(3)} \]

### Step 8: Convert \( \log_{4}(9) \)
We know that \( 9 = 3^2 \), so:
\[ \log_{4}(9) = 2\log_{4}(3) \]
Thus, substituting back gives:
\[ \log_{4}x = \frac{2\log_{4}(3)}{\log_{4}(3)} = 2 \]

### Step 9: Solve for \( x \)
Taking antilogarithm gives:
\[ x = 4^2 = 16 \]

### Final Answer
The solution to the equation \( x^{\log_{4}3} + 3^{\log_{4}x} = 18 \) is:
\[ \boxed{16} \]