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

Question

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using the Quotient Rule of Logarithms, the Product Rule of Logarithms, and the Power Rule of Logarithms?

Option #1: log49−3log4m+log4n
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Option #2: log49+3log4m−log4n
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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: \[ \log_4(9) + 3\log_4(m) + \log_4(n) \]