Which of the following options correctly rewrites log4(9m3n)

Question

using the Quotient Rule of Logarithms, the Product Rule of Logarithms, and the Power Rule of Logarithms?
only answers i gaev

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 logarithmic rules, we can break it down using the Product Rule and the Power Rule.

1. Product Rule: \( \log_b(xy) = \log_b(x) + \log_b(y) \)

   • Here, \( \log_4(9m^3n) = \log_4(9) + \log_4(m^3) + \log_4(n) \)

2. Power Rule: \( \log_b(x^k) = k \log_b(x) \)

   • Hence, \( \log_4(m^3) = 3\log_4(m) \)
   • So now we have \( \log_4(9) + 3\log_4(m) + \log_4(n) \)

This means we can write: \[ \log_4(9m^3n) = \log_4(9) + 3\log_4(m) + \log_4(n) \]

Now, we need to verify if any of the given options match this expression:

• Option #1: \( \log_4(9) - 3\log_4(m) + \log_4(n) \) (Incorrect)

• Option #2: \( \log_4(9) + 3\log_4(m) - \log_4(n) \) (Incorrect)

• Option #3: \( 3\log_4(9) - \log_4(n) \) (Incorrect)

None of the options provided directly match \( \log_4(9) + 3\log_4(m) + \log_4(n) \).

Based on my analysis, although none of the provided options correspond correctly, if I must choose one that one aligns the closest without verification of signs, I would suggest Option #2 as it has positive \( \log_4(9) \), but it has the incorrect sign with \( \log_4(n) \).

In conclusion, none of the options accurately represent the original expression, but if forced to pick one based on the workings above, I would select option #2.