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 will use the properties of logarithms.

### Step 1: Express x in terms of log a
Given:
\[ 2 \log_{8} a = x \]

We can convert the logarithm to base 8 into base 2:
\[ \log_{8} a = \frac{\log_{2} a}{\log_{2} 8} \]

Since \( \log_{2} 8 = 3 \) (because \( 8 = 2^3 \)), we have:
\[ \log_{8} a = \frac{\log_{2} a}{3} \]

Now substituting this back into the equation for x:
\[ x = 2 \cdot \frac{\log_{2} a}{3} = \frac{2}{3} \log_{2} a \]

### Step 2: Express y in terms of log a
Next, we have:
\[ \log_{2} (2a) = y \]

Using the property of logarithms:
\[ \log_{2} (2a) = \log_{2} 2 + \log_{2} a = 1 + \log_{2} a \]

So, we can express y as:
\[ y = 1 + \log_{2} a \]

### Step 3: Set up the equation using y - x = 4
We know from the problem statement that:
\[ y - x = 4 \]

Substituting the expressions for y and x:
\[ (1 + \log_{2} a) - \left(\frac{2}{3} \log_{2} a\right) = 4 \]

### Step 4: Simplify the equation
Now, we simplify the left-hand side:
\[ 1 + \log_{2} a - \frac{2}{3} \log_{2} a = 4 \]

Combining the logarithmic terms:
\[ 1 + \left(1 - \frac{2}{3}\right) \log_{2} a = 4 \]
\[ 1 + \frac{1}{3} \log_{2} a = 4 \]

### Step 5: Solve for log a
Subtract 1 from both sides:
\[ \frac{1}{3} \log_{2} a = 3 \]

Now, multiply both sides by 3:
\[ \log_{2} a = 9 \]

### Step 6: Find the value of a
To find a, we can rewrite the logarithmic equation:
\[ a = 2^{9} = 512 \]

### Step 7: Substitute back to find x
Now substitute \( \log_{2} a \) back into the equation for x:
\[ x = \frac{2}{3} \log_{2} a = \frac{2}{3} \cdot 9 = 6 \]

Thus, the value of x is:
\[ \boxed{6} \]