If `2x^(log_(4)3)+3^(log_(4)x)= 27`, then x is equal to

Options

A. 2
B. 4
C. 8
D. 16

Explanation

To solve the equation \( 2x^{\log_{4}3} + 3^{\log_{4}x} = 27 \), we will follow these steps:

### Step 1: Rewrite the logarithmic expressions
We know that \( \log_{a}b = \frac{\log_{c}b}{\log_{c}a} \). We can rewrite \( \log_{4}3 \) and \( \log_{4}x \) using base 10 or base e (natural logarithm). However, for simplicity, we will keep them as they are for now.

### Step 2: Substitute \( y = \log_{4}x \)
Let \( y = \log_{4}x \). Then, we can express \( x \) in terms of \( y \):
\[ x = 4^y \]

### Step 3: Substitute \( x \) into the equation
Now, substitute \( x = 4^y \) into the original equation:
\[ 2(4^y)^{\log_{4}3} + 3^{y} = 27 \]

### Step 4: Simplify the equation
Using the property of exponents, we can simplify \( (4^y)^{\log_{4}3} \) as follows:
\[ (4^y)^{\log_{4}3} = 4^{y \cdot \log_{4}3} = 3^y \]
Thus, the equation becomes:
\[ 2 \cdot 3^y + 3^y = 27 \]
This simplifies to:
\[ 3 \cdot 3^y = 27 \]

### Step 5: Solve for \( y \)
Now, divide both sides by 3:
\[ 3^y = 9 \]
Since \( 9 = 3^2 \), we have:
\[ y = 2 \]

### Step 6: Substitute back to find \( x \)
Recall that \( y = \log_{4}x \). Therefore:
\[ \log_{4}x = 2 \]
This implies:
\[ x = 4^2 = 16 \]

### Final Answer
Thus, the value of \( x \) is:
\[ \boxed{16} \]