If `2"log"_(8) a =x, "log"_(2) 2a = y " and " y-x =4,` then x =

Options

A. 10
B. 16
C. 4
D. 6

Answer

To solve the problem step by step, we start with the given equations:

1. \( 2 \log_8 a = x \)
2. \( \log_2 2a = y \)
3. \( y - x = 4 \)

### Step 1: Rewrite the logarithm in terms of base 2

Using the change of base formula, we can rewrite \( \log_8 a \) in terms of base 2:

\[ \log_8 a = \frac{\log_2 a}{\log_2 8} \]

Since \( 8 = 2^3 \), we have \( \log_2 8 = 3 \). Therefore:

\[ \log_8 a = \frac{\log_2 a}{3} \]

Substituting this back into the equation for \( x \):

\[ x = 2 \log_8 a = 2 \cdot \frac{\log_2 a}{3} = \frac{2}{3} \log_2 a \]

### Step 2: Rewrite \( y \)

Now, we can rewrite \( y \):

\[ y = \log_2 2a = \log_2 2 + \log_2 a \]

Since \( \log_2 2 = 1 \), we have:

\[ y = 1 + \log_2 a \]

### Step 3: Substitute \( x \) and \( y \) into the third equation

Now substitute \( x \) and \( y \) into the equation \( y - x = 4 \):

\[ (1 + \log_2 a) - \left(\frac{2}{3} \log_2 a\right) = 4 \]

### Step 4: Simplify the equation

Combine the terms involving \( \log_2 a \):

\[ 1 + \log_2 a - \frac{2}{3} \log_2 a = 4 \]

This simplifies to:

\[ 1 + \left(1 - \frac{2}{3}\right) \log_2 a = 4 \]

\[ 1 + \frac{1}{3} \log_2 a = 4 \]

### Step 5: Isolate \( \log_2 a \)

Subtract 1 from both sides:

\[ \frac{1}{3} \log_2 a = 3 \]

Multiply both sides by 3:

\[ \log_2 a = 9 \]

### Step 6: Find \( x \)

Now substitute \( \log_2 a = 9 \) back into the equation for \( x \):

\[ x = \frac{2}{3} \log_2 a = \frac{2}{3} \cdot 9 = 6 \]

### Final Answer

Thus, the value of \( x \) is:

\[ \boxed{6} \]