To rewrite \( \log_4(9m^3n) \) using the rules of logarithms (Product Rule, Quotient Rule, and Power Rule), let's break it down step by step.
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Use the Product Rule: The Product Rule states that \( \log_b(xy) = \log_b(x) + \log_b(y) \). Here, we can apply it to the three parts of \( 9m^3n \): \[ \log_4(9m^3n) = \log_4(9) + \log_4(m^3) + \log_4(n) \]
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Use the Power Rule: The Power Rule states that \( \log_b(x^k) = k \cdot \log_b(x) \). Here, we can apply it to \( m^3 \): \[ \log_4(m^3) = 3 \log_4(m) \]
Now we can rewrite the expression: \[ \log_4(9m^3n) = \log_4(9) + 3 \log_4(m) + \log_4(n) \]
This rearrangement gives us: \[ \log_4(9) + 3\log_4(m) + \log_4(n) \]
Now we can match this with the options given:
- Option #1: \( \log_4(9) - 3\log_4(m) + \log_4(n) \) (incorrect sign for the \( \log_4(m) \) term)
- Option #2: \( \log_4(9) + 3\log_4(m) - \log_4(n) \) (incorrect sign for the \( \log_4(n) \) term)
- Option #3: \( 3\log_4(9m) - \log_4(n) \) (wrong format)
None of the options correctly match our derived expression exactly. However, if the question required considering \( \log_4(9) \) to take part in a calculation with 3 and \( m \) in a different way, we could explore options again.
From the analysis, you should look for a log property/application that might adjust this or clarify, as the main calculation yields:
Final Expression: \[ \log_4(9) + 3\log_4(m) + \log_4(n) \]
Check wording if specific substitutions applied lead to \( log_4(9m^3) - log_4(n) \) might result in a plausible log equivalence of offered options. Thus would answer determining that per checks here none strictly adherent to ideal simplification, although Option #3 deviates. Results may suggest misalignment on provided concepts.