Let's evaluate the expression:
[tex]\[
\frac{\log 6^5 - \log 5^6}{\log 6^{\frac{1}{5}} + \log 5^6}
\][/tex]
### Step-by-step Solution:
1. Simplify the terms involving logarithms:
- Recall the properties of logarithms: [tex]\(\log a^b = b \log a\)[/tex].
- Apply this property to each term in the numerator and denominator.
2. Simplify the numerator:
[tex]\[
\log 6^5 - \log 5^6 = 5 \log 6 - 6 \log 5
\][/tex]
3. Simplify the denominator:
[tex]\[
\log 6^{\frac{1}{5}} + \log 5^6 = \frac{1}{5} \log 6 + 6 \log 5
\][/tex]
4. Combine the simplified terms:
The fraction now looks like:
[tex]\[
\frac{5 \log 6 - 6 \log 5}{\frac{1}{5} \log 6 + 6 \log 5}
\][/tex]
5. Evaluate the simplified form:
- Use the given logarithmic values and respective calculations.
- Numerator: [tex]\( 5 \log 6 - 6 \log 5 = -0.6978301284643287 \)[/tex]
- Denominator: [tex]\( \frac{1}{5} \log 6 + 6 \log 5 = 10.014979368450215 \)[/tex]
6. Calculate the final value of the expression:
[tex]\[
\frac{-0.6978301284643287}{10.014979368450215} = -0.06967863864629364
\][/tex]
### Conclusion:
Thus, the value of the expression is:
[tex]\[
\boxed{-0.06967863864629364}
\][/tex]
