Which of the following options correctly rewrites log4(9m^3/n)

Question

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.