Let's solve the given expression step by step:

The expression is: $\log(35) - \log(5) + \log\left(\frac{7}{8}\right)$

Step 1: Use the logarithmic subtraction property

$\log(a) - \log(b) = \log\left(\frac{a}{b}\right)$ Applying this to the first two terms: $\log(35) - \log(5) = \log\left(\frac{35}{5}\right) = \log(7)$

Step 2: Use the logarithmic addition property

$\log(a) + \log(b) = \log(ab)$ Now add the remaining term: $\log(7) + \log\left(\frac{7}{8}\right) = \log\left(7 \times \frac{7}{8}\right) = \log\left(\frac{49}{8}\right)$

Final Expression

The simplified expression is: $\log\left(\frac{49}{8}\right)$

Looking at the answer choices, we see that this corresponds to option D: $\log(7^2) + \log(8) = \log\left(\frac{49}{8}\right)$

Would you like more details or have any further questions?

Here are five related questions to explore this topic further:

1. How does the property $\log(a) - \log(b) = \log\left(\frac{a}{b}\right)$ work in general?
2. Can you simplify $\log\left(\frac{x}{y}\right)$ using logarithmic properties?
3. What happens when you apply the power rule $\log(a^b) = b\log(a)$?
4. How would you solve similar problems with natural logarithms $\ln$ instead of common logarithms $\log$?
5. Can the logarithmic properties be applied to non-real numbers?

Tip: Always remember that the logarithmic rules can simplify complex expressions, but they must be applied correctly in the right order to avoid mistakes.