Question

Which of the following options correctly rewrites log4(9m3n)
log
4
⁡
(
9
𝑚
3
𝑛
)
using the Quotient Rule of Logarithms, the Product Rule of Logarithms, and the Power Rule of Logarithms?

Option #1: log49−3log4m+log4n
log
4
⁡
9
−
3
log
4
⁡
𝑚
+
log
4
⁡
𝑛

Option #2: log49+3log4m−log4n
log
4
⁡
9
+
3
log
4
⁡
𝑚
−
log
4
⁡
𝑛

Option #3: 3log49m−log4n

Answer 1

To rewrite \( \log_4(9m^3n) \) using the Quotient Rule, Product Rule, and Power Rule of Logarithms, we can follow these steps:

1. **Product Rule:** The logarithm of a product is the sum of the logarithms:
\[
\log_b(xy) = \log_b(x) + \log_b(y)
\]
Therefore, we can apply this rule to separate \( 9 \), \( m^3 \), and \( n \):
\[
\log_4(9m^3n) = \log_4(9) + \log_4(m^3) + \log_4(n)
\]

2. **Power Rule:** The logarithm of a power is the exponent times the logarithm of the base:
\[
\log_b(x^k) = k \log_b(x)
\]
Applying this rule to \( \log_4(m^3) \):
\[
\log_4(m^3) = 3\log_4(m)
\]
Now we can write the expression as:
\[
\log_4(9) + 3\log_4(m) + \log_4(n)
\]

Putting it all together, we have:
\[
\log_4(9m^3n) = \log_4(9) + 3\log_4(m) + \log_4(n)
\]

Now we can compare this with the provided options:

- **Option #1:** \( \log_4(9) - 3\log_4(m) + \log_4(n) \)
- **Option #2:** \( \log_4(9) + 3\log_4(m) - \log_4(n) \)
- **Option #3:** \( 3\log_4(9) - \log_4(n) \)

None of the options perfectly match our derived formula. However, the closest match in structure to the derived expression (\( \log_4(9) + 3\log_4(m) + \log_4(n) \)) appears to be **Option #2**, but it has a subtraction with \( \log_4(n) \) which is incorrect.

Based on the breakdown, none of the options are correct according to the rules of logarithms. The correct expression should be:
\[
\log_4(9) + 3\log_4(m) + \log_4(n)
\]