Question

Which of the following options correctly rewrites log4(9m^3/n)
using the Quotient Rule of Logarithms, the Product Rule of Logarithms, and the Power Rule of Logarithms?

Option #1: log4 9−3 log4 m+log4 n


Option #2: log4 9+3 log4 m−log4 n


Option #3: 3 log4 9m−log4 n


(1 point)
Option #

Answer 1

To rewrite \( \log_4\left(\frac{9m^3}{n}\right) \) using the properties of logarithms, we will apply the Quotient Rule, Product Rule, and Power Rule as follows:

1. **Quotient Rule**: \( \log_b\left(\frac{A}{B}\right) = \log_b A - \log_b B \)
- In this case, \( A = 9m^3 \) and \( B = n \).
- So, \( \log_4\left(\frac{9m^3}{n}\right) = \log_4(9m^3) - \log_4(n) \).

2. **Product Rule**: \( \log_b(AB) = \log_b A + \log_b B \)
- Now applying this to \( \log_4(9m^3) \) where \( A = 9 \) and \( B = m^3 \).
- This gives us \( \log_4(9m^3) = \log_4(9) + \log_4(m^3) \).

3. **Power Rule**: \( \log_b(A^c) = c \cdot \log_b A \)
- For \( \log_4(m^3) \), we can rewrite it as \( 3 \cdot \log_4(m) \).
- Therefore, \( \log_4(9m^3) = \log_4(9) + 3 \cdot \log_4(m) \).

Combining everything together, we have:

\[
\log_4\left(\frac{9m^3}{n}\right) = \left(\log_4(9) + 3 \cdot \log_4(m)\right) - \log_4(n)
\]

Now simplifying this, we get:

\[
\log_4(9) + 3 \cdot \log_4(m) - \log_4(n)
\]

This matches with **Option #2**:

\[
\log_4(9) + 3 \cdot \log_4(m) - \log_4(n)
\]

So the correct answer is:

**Option #2**.