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

Options

A. 502
B. 512
C. 416
D. none of these

Answer

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

1. **Given Equations:**
- \( 2 \log_{8} N = p \)
- \( \log_{2} (2N) = q \)
- \( q - p = 4 \)

2. **Convert the logarithm base:**
- We can convert \( \log_{8} N \) to base 2 using the change of base formula:
\[ \log_{8} N = \frac{\log_{2} N}{\log_{2} 8} \]
- Since \( \log_{2} 8 = 3 \) (because \( 8 = 2^3 \)), we have:
\[ \log_{8} N = \frac{\log_{2} N}{3} \]
- Therefore, substituting this into the first equation:
\[ 2 \log_{8} N = 2 \cdot \frac{\log_{2} N}{3} = \frac{2 \log_{2} N}{3} \]
- Thus, we can rewrite \( p \) as:
\[ p = \frac{2 \log_{2} N}{3} \]

3. **Simplifying \( q \):**
- For \( q \):
\[ q = \log_{2} (2N) = \log_{2} 2 + \log_{2} N = 1 + \log_{2} N \]

4. **Substituting into the equation \( q - p = 4 \):**
- Now, substituting \( p \) and \( q \) into the equation \( q - p = 4 \):
\[ (1 + \log_{2} N) - \left(\frac{2 \log_{2} N}{3}\right) = 4 \]

5. **Combining the terms:**
- To combine the terms, we need a common denominator:
\[ 1 + \log_{2} N - \frac{2 \log_{2} N}{3} = 4 \]
- Rewrite \( \log_{2} N \) with a common denominator:
\[ 1 + \frac{3 \log_{2} N}{3} - \frac{2 \log_{2} N}{3} = 4 \]
- This simplifies to:
\[ 1 + \frac{1 \log_{2} N}{3} = 4 \]

6. **Isolating \( \log_{2} N \):**
- Subtract 1 from both sides:
\[ \frac{1 \log_{2} N}{3} = 3 \]
- Multiply both sides by 3:
\[ \log_{2} N = 9 \]

7. **Finding \( N \):**
- To find \( N \), we convert back from logarithmic form:
\[ N = 2^9 = 512 \]

Thus, the value of \( N \) is \( 512 \).