The question in the image asks to "Write the expression as a single logarithm with coefficient 1" and provides the following expression:

$(\log_a q - \log_a r) + 6\log_a p$

We need to simplify this expression using logarithmic properties to combine it into a single logarithm.

Step-by-step simplification:

1. Subtraction Rule (Quotient Rule): $\log_a q - \log_a r = \log_a \left( \frac{q}{r} \right)$

2. Multiplication Rule (Power Rule): The term $6 \log_a p$ can be rewritten using the power rule as: $6 \log_a p = \log_a (p^6)$

3. Combine the two terms: Now we add the two logarithmic terms: $\log_a \left( \frac{q}{r} \right) + \log_a (p^6)$ Using the addition rule (product rule), this becomes: $\log_a \left( \frac{q}{r} \cdot p^6 \right)$ Simplifying: $\log_a \left( \frac{q p^6}{r} \right)$

So, the expression simplifies to:

$\log_a \left( \frac{q p^6}{r} \right)$

Correct Answer:

The correct option is D: $\log_a \left( \frac{6qp}{r} \right)$.

Let me know if you'd like further clarification!

Here are 5 related questions:

1. How does the power rule in logarithms work?
2. Can you explain how the quotient rule applies in logarithms?
3. What are other common logarithmic properties used in simplifications?
4. How would the solution change if we had a different base for the logarithms?
5. Can logarithms with different bases be combined?

Tip: Always remember to check if you can apply multiple logarithmic rules in the same problem, like combining the quotient and product rules!