1. The problem is to simplify the expression $\log \frac{41}{35} + \log 70 - \log \frac{41}{2} + \log 5^2$. 2. Use the logarithm property $\log a + \log b = \log (a \times b)$ and $\log a - \log b = \log \frac{a}{b}$ to combine terms. 3. Combine the first two terms: $$\log \frac{41}{35} + \log 70 = \log \left( \frac{41}{35} \times 70 \right) = \log \left( \frac{41 \times 70}{35} \right)$$ 4. Simplify inside the logarithm: $$\frac{41 \times 70}{35} = 41 \times 2 = 82$$ 5. Now the expression is: $$\log 82 - \log \frac{41}{2} + \log 5^2$$ 6. Use the subtraction property: $$\log 82 - \log \frac{41}{2} = \log \left( \frac{82}{\frac{41}{2}} \right) = \log \left( 82 \times \frac{2}{41} \right)$$ 7. Simplify inside the logarithm: $$82 \times \frac{2}{41} = 2 \times 2 = 4$$ 8. Now the expression is: $$\log 4 + \log 5^2$$ 9. Use the property $\log a + \log b = \log (a \times b)$: $$\log 4 + \log 25 = \log (4 \times 25) = \log 100$$ 10. Since $\log 100 = 2$ (assuming base 10), the simplified value is: $$2$$ Final answer: $2$