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