To simplify the expression \( \log_4 64 - \log_4 16 \) using the Quotient Rule of Logarithms, we will follow these steps:
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Apply the Quotient Rule: The quotient rule states that \( \log_b a - \log_b c = \log_b \left( \frac{a}{c} \right) \). In our case, we have: \[ \log_4 64 - \log_4 16 = \log_4 \left( \frac{64}{16} \right) \]
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Simplify the fraction: Next, we simplify \( \frac{64}{16} \): \[ \frac{64}{16} = 4 \]
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Substitute back into the logarithm: Now we substitute back into our logarithm: \[ \log_4 \left( \frac{64}{16} \right) = \log_4 4 \]
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Evaluate the logarithm: We know that \( \log_4 4 = 1 \) because \( 4^1 = 4 \).
Thus, the simplified expression is: \[ \log_4 64 - \log_4 16 = 1 \]
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