Question

Which of the following options correctly rewrites log4(9m3n)
using the Quotient Rule of Logarithms, the Product Rule of Logarithms, and the Power Rule of Logarithms?

Option #1: log49−3log4m+log4n

Option #2: log49+3log4m−log4n

Option #3: 3log49m−log4n

(1 point)
Option #

Answer 1

To rewrite \( \log_4(9m^3n) \) using the rules of logarithms (Product Rule, Quotient Rule, and Power Rule), let's break it down step by step.

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

2. **Use the Power Rule**: The Power Rule states that \( \log_b(x^k) = k \cdot \log_b(x) \). Here, we can apply it to \( m^3 \):
\[
\log_4(m^3) = 3 \log_4(m)
\]

Now we can rewrite the expression:
\[
\log_4(9m^3n) = \log_4(9) + 3 \log_4(m) + \log_4(n)
\]

This rearrangement gives us:
\[
\log_4(9) + 3\log_4(m) + \log_4(n)
\]

Now we can match this with the options given:

- **Option #1**: \( \log_4(9) - 3\log_4(m) + \log_4(n) \) (incorrect sign for the \( \log_4(m) \) term)
- **Option #2**: \( \log_4(9) + 3\log_4(m) - \log_4(n) \) (incorrect sign for the \( \log_4(n) \) term)
- **Option #3**: \( 3\log_4(9m) - \log_4(n) \) (wrong format)

None of the options correctly match our derived expression exactly. However, if the question required considering \( \log_4(9) \) to take part in a calculation with 3 and \( m \) in a different way, we could explore options again.

From the analysis, you should look for a log property/application that might adjust this or clarify, as the main calculation yields:

**Final Expression**:
\[
\log_4(9) + 3\log_4(m) + \log_4(n)
\]

Check wording if specific substitutions applied lead to \( log_4(9m^3) - log_4(n) \) might result in a plausible log equivalence of offered options. Thus would answer determining that per checks here none strictly adherent to ideal simplification, although Option #3 deviates. Results may suggest misalignment on provided concepts.