If `"log"_(4) 2 + "log"_(4) 4 + "log"_(4) 16 + "log"_(4) x = 6`, then x =

Options

A. 4
B. 64
C. 32
D. 8

Answer

To solve the equation \( \log_{4} 2 + \log_{4} 4 + \log_{4} 16 + \log_{4} x = 6 \), we can follow these steps:

### Step 1: Simplify the logarithmic terms
Using the property of logarithms that states \( \log_{b} a + \log_{b} c = \log_{b} (a \cdot c) \), we can combine the logarithmic terms on the left-hand side.

\[ \log_{4} 2 + \log_{4} 4 + \log_{4} 16 + \log_{4} x = \log_{4} (2 \cdot 4 \cdot 16 \cdot x) \]

### Step 2: Evaluate \( \log_{4} 4 \)
We know that \( \log_{4} 4 = 1 \). Thus, we can rewrite the equation as:

\[ \log_{4} (2 \cdot 4 \cdot 16 \cdot x) = 6 \]

### Step 3: Rewrite the equation
This means:

\[ \log_{4} (2 \cdot 4 \cdot 16 \cdot x) = \log_{4} (4^6) \]

### Step 4: Set the arguments equal
Since the logarithms are equal, we can set the arguments equal to each other:

\[ 2 \cdot 4 \cdot 16 \cdot x = 4^6 \]

### Step 5: Calculate \( 4^6 \)
We know that \( 4^6 = (2^2)^6 = 2^{12} \).

### Step 6: Calculate the left-hand side
Now, calculate \( 2 \cdot 4 \cdot 16 \):

\[ 2 \cdot 4 = 8 \]
\[ 8 \cdot 16 = 128 \]

So, we have:

\[ 128 \cdot x = 4^6 \]

### Step 7: Substitute \( 4^6 \)
Substituting \( 4^6 \) gives us:

\[ 128 \cdot x = 4096 \]

### Step 8: Solve for \( x \)
Now, divide both sides by 128:

\[ x = \frac{4096}{128} \]

Calculating this gives:

\[ x = 32 \]

### Final Answer
Thus, the value of \( x \) is \( 32 \).

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