If p = `log_(3)5` and `q= log_(17) 25` which one of the following is correct?

Options

A. `p lt q`
B. p =q
C. `p gt q`
D. can't say

Answer

To solve the problem, we need to compare the values of \( p = \log_3 5 \) and \( q = \log_{17} 25 \).

### Step-by-Step Solution:

1. **Convert the logarithms to a common base**:
We can use the change of base formula for logarithms, which states that:
\[ \log_a b = \frac{\log_c b}{\log_c a} \]
We will convert both \( p \) and \( q \) to base 10 (or any common base).

For \( p \):
\[ p = \log_3 5 = \frac{\log_{10} 5}{\log_{10} 3} \]

For \( q \):
\[ q = \log_{17} 25 = \frac{\log_{10} 25}{\log_{10} 17} \]

2. **Simplify \( \log_{10} 25 \)**:
We know that \( 25 = 5^2 \), so:
\[ \log_{10} 25 = \log_{10} (5^2) = 2 \log_{10} 5 \]
Therefore, we can rewrite \( q \) as:
\[ q = \frac{2 \log_{10} 5}{\log_{10} 17} \]

3. **Now we have expressions for both \( p \) and \( q \)**:
\[ p = \frac{\log_{10} 5}{\log_{10} 3} \]
\[ q = \frac{2 \log_{10} 5}{\log_{10} 17} \]

4. **Compare \( p \) and \( q \)**:
To compare \( p \) and \( q \), we can set up the inequality:
\[ p > q \quad \text{if} \quad \frac{\log_{10} 5}{\log_{10} 3} > \frac{2 \log_{10} 5}{\log_{10} 17} \]
This simplifies to:
\[ \log_{10} 5 \cdot \log_{10} 17 > 2 \log_{10} 3 \cdot \log_{10} 5 \]
Since \( \log_{10} 5 \) is positive, we can divide both sides by \( \log_{10} 5 \):
\[ \log_{10} 17 > 2 \log_{10} 3 \]

5. **Evaluate \( \log_{10} 17 \) and \( 2 \log_{10} 3 \)**:
Using approximate values:
- \( \log_{10} 3 \approx 0.477 \)
- Therefore, \( 2 \log_{10} 3 \approx 0.954 \)
- \( \log_{10} 17 \approx 1.230 \)

Since \( 1.230 > 0.954 \), we conclude that:
\[ \log_{10} 17 > 2 \log_{10} 3 \]

6. **Final conclusion**:
Since we have established that \( p > q \), the correct option is:
\[ p \text{ is greater than } q \]

### Answer:
**Option 3: \( p > q \)**