Question

Which of the following options correctly rewrites
log
4
(
9
m
3
n
)
using the Quotient Rule of Logarithms, the Product Rule of Logarithms, and the Power Rule of Logarithms?

Option #1:
log
4
9
−
3
log
4
m
+
log
4
n


Option #2:
log
4
9
+
3
log
4
m
−
log
4
n


Option #3:
3
log
4
9
m
−
log
4
n


(1 point)

Answer 1

To rewrite the expression \( \log_4(9m^3n) \) using the Quotient Rule, Product Rule, and Power Rule of Logarithms, we start by applying these rules step-by-step.

1. **Product Rule**: The Product Rule states that \( \log_b(xy) = \log_b(x) + \log_b(y) \).
- Here, we can apply this to the product \( 9m^3n \):
\[
\log_4(9m^3n) = \log_4(9) + \log_4(m^3) + \log_4(n)
\]

2. **Power Rule**: The Power Rule states that \( \log_b(x^k) = k \cdot \log_b(x) \).
- Applying the Power Rule to \( m^3 \):
\[
\log_4(m^3) = 3 \cdot \log_4(m)
\]

Putting everything together:
\[
\log_4(9m^3n) = \log_4(9) + 3\log_4(m) + \log_4(n)
\]

This expression matches **Option #2**:
\[
\log_4 9 + 3 \log_4 m + \log_4 n
\]

In your question, we also notice the signs of the logarithms in the options provided. The correct formulation is:
\[
\log_4(9) + 3\log_4(m) + \log_4(n)
\]
Thus, **Option #2** is incorrect when compared to this analysis since it suggests a subtraction. Each term is added.

The only correct interpretation from the elaboration is conceptual, leading to verify if other options seem trivial.

### Thus, looking at all options:
- **Option #1**: Incorrect due to subtraction.
- **Option #2**: Incorrect sign.
- **Option #3**: Incorrect structure.

In conclusion, none of the provided options is formally correct based on logarithmic identities—highlighting a clear review for clarity.

If you wish for a reiteration, the structure properly on \( \log_4(9m^3n) \) should remain validated and emerge as two addended terms correctly positioned.