If `2log_(8)N = p, log_(2)N = q and q - p = 4` then the value of N is

Options

A. 4096
B. 256
C. 512
D. 32

Answer

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

1. \( 2 \log_8 N = p \)
2. \( \log_2 N = q \)
3. \( q - p = 4 \)

### Step 1: Express \( p \) in terms of \( q \)

From the third equation, we can express \( p \) as:
\[ p = q - 4 \]

### Step 2: Rewrite \( p \) using the change of base formula

Using the change of base formula for logarithms, we can rewrite \( \log_8 N \):
\[ \log_8 N = \frac{\log_2 N}{\log_2 8} \]
Since \( \log_2 8 = 3 \) (because \( 8 = 2^3 \)), we have:
\[ \log_8 N = \frac{\log_2 N}{3} = \frac{q}{3} \]

Now substituting this into the expression for \( p \):
\[ p = 2 \log_8 N = 2 \cdot \frac{q}{3} = \frac{2q}{3} \]

### Step 3: Substitute \( p \) into the equation \( q - p = 4 \)

Now we substitute \( p \) into the equation:
\[ q - \frac{2q}{3} = 4 \]

### Step 4: Simplify the equation

To simplify, we can find a common denominator:
\[ \frac{3q}{3} - \frac{2q}{3} = 4 \]
This simplifies to:
\[ \frac{q}{3} = 4 \]

### Step 5: Solve for \( q \)

Multiplying both sides by 3 gives:
\[ q = 12 \]

### Step 6: Substitute \( q \) back to find \( N \)

Now we know \( q = \log_2 N \), so we can write:
\[ \log_2 N = 12 \]

Using the definition of logarithms, we can express \( N \) as:
\[ N = 2^{12} \]

### Step 7: Calculate \( N \)

Calculating \( 2^{12} \):
\[ N = 4096 \]

Thus, the value of \( N \) is:
\[ \boxed{4096} \]

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