If = `log "" ( 5)/(7) , m = log ""( 7)/(9) and n = 2 ( log 3 - log sqrt5)` , find the value of
` 7^(l+ m + n ) `

Answer

To solve the problem, we need to find the value of \( 7^{L + M + N} \) where \( L = \log \frac{5}{7} \), \( M = \log \frac{7}{9} \), and \( N = 2(\log 3 - \log \sqrt{5}) \).

### Step-by-Step Solution:

1. **Write down the given values**:
\[ L = \log \frac{5}{7}, \quad M = \log \frac{7}{9}, \quad N = 2(\log 3 - \log \sqrt{5}) \]

2. **Simplify \( N \)**:
\[ N = 2(\log 3 - \log \sqrt{5}) = 2(\log 3 - \log 5^{1/2}) = 2(\log 3 - \frac{1}{2} \log 5) \]
Using the property \( c \cdot \log a = \log a^c \):
\[ N = 2\log 3 - \log 5 = \log 3^2 - \log 5 = \log 9 - \log 5 \]
Using the property \( \log a - \log b = \log \frac{a}{b} \):
\[ N = \log \frac{9}{5} \]

3. **Combine \( L + M + N \)**:
\[ L + M + N = \log \frac{5}{7} + \log \frac{7}{9} + \log \frac{9}{5} \]
Using the property \( \log a + \log b = \log(ab) \):
\[ L + M + N = \log \left( \frac{5}{7} \cdot \frac{7}{9} \cdot \frac{9}{5} \right) \]

4. **Simplify the expression inside the logarithm**:
\[ \frac{5}{7} \cdot \frac{7}{9} \cdot \frac{9}{5} = \frac{5 \cdot 7 \cdot 9}{7 \cdot 9 \cdot 5} = 1 \]
Thus,
\[ L + M + N = \log 1 \]

5. **Evaluate \( \log 1 \)**:
\[ \log 1 = 0 \]

6. **Find \( 7^{L + M + N} \)**:
\[ 7^{L + M + N} = 7^0 = 1 \]

### Final Answer:
\[ \boxed{1} \]