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

Answer

To solve the problem, we need to find the value of \( l + m + n \) where:

- \( l = \log \frac{5}{7} \)
- \( m = \log \frac{7}{9} \)
- \( n = 2(\log 3 - \log \sqrt{5}) \)

### Step-by-Step Solution:

1. **Write down the expressions for \( l \), \( m \), and \( n \)**:
\[ l = \log \frac{5}{7}, \quad m = \log \frac{7}{9}, \quad n = 2(\log 3 - \log \sqrt{5}) \]

2. **Combine \( l \) and \( m \)**:
Using the property of logarithms that states \( \log a + \log b = \log(ab) \):
\[ l + m = \log \frac{5}{7} + \log \frac{7}{9} = \log \left( \frac{5}{7} \cdot \frac{7}{9} \right) \]
The \( 7 \) in the numerator and denominator cancels out:
\[ l + m = \log \frac{5}{9} \]

3. **Simplify \( n \)**:
We know that \( \log \sqrt{5} = \frac{1}{2} \log 5 \), thus:
\[ n = 2(\log 3 - \log \sqrt{5}) = 2\left(\log 3 - \frac{1}{2} \log 5\right) = 2\log 3 - \log 5 \]

4. **Combine \( l + m + n \)**:
Now, we add \( n \) to \( l + m \):
\[ l + m + n = \log \frac{5}{9} + (2\log 3 - \log 5) \]
We can rewrite \( 2\log 3 \) as \( \log 3^2 \):
\[ l + m + n = \log \frac{5}{9} + \log 9 - \log 5 \]

5. **Combine the logarithmic terms**:
Using the property \( \log a - \log b = \log \left( \frac{a}{b} \right) \):
\[ l + m + n = \log \frac{5 \cdot 9}{9 \cdot 5} = \log 1 \]

6. **Final result**:
Since \( \log 1 = 0 \):
\[ l + m + n = 0 \]

### Final Answer:
\[ l + m + n = 0 \]