Solve the following equations.
(iii) `2.x^(log_(4)3)+3^(log_4x)=27`

Answer

To solve the equation \( 2 \cdot x^{\log_4 3} + 3^{\log_4 x} = 27 \), we can follow these steps:

### Step 1: Rewrite the equation
We start with the equation:
\[ 2 \cdot x^{\log_4 3} + 3^{\log_4 x} = 27 \]

### Step 2: Use the property of logarithms
Using the property \( a^{\log_b c} = c^{\log_b a} \), we can rewrite \( 3^{\log_4 x} \):
\[ 3^{\log_4 x} = x^{\log_4 3} \]
Thus, our equation becomes:
\[ 2 \cdot x^{\log_4 3} + x^{\log_4 3} = 27 \]

### Step 3: Combine like terms
Now, we can combine the terms:
\[ (2 + 1) \cdot x^{\log_4 3} = 27 \]
This simplifies to:
\[ 3 \cdot x^{\log_4 3} = 27 \]

### Step 4: Isolate \( x^{\log_4 3} \)
Next, we divide both sides by 3:
\[ x^{\log_4 3} = \frac{27}{3} = 9 \]

### Step 5: Express 9 as a power of 3
We know that \( 9 = 3^2 \), so we can write:
\[ x^{\log_4 3} = 3^2 \]

### Step 6: Take logarithm base 4
Taking logarithm base 4 of both sides gives us:
\[ \log_4 (x^{\log_4 3}) = \log_4 (3^2) \]

### Step 7: Apply the power rule of logarithms
Using the power rule of logarithms, we have:
\[ \log_4 3 \cdot \log_4 x = 2 \cdot \log_4 3 \]

### Step 8: Divide by \( \log_4 3 \)
Assuming \( \log_4 3 \neq 0 \), we can divide both sides by \( \log_4 3 \):
\[ \log_4 x = 2 \]

### Step 9: Convert back to exponential form
Now, converting back from logarithmic form gives us:
\[ x = 4^2 = 16 \]

### Final Answer
Thus, the solution to the equation is:
\[ \boxed{16} \]