To demonstrate that the Quotient Rule of Logarithms holds for the base 4 expression \( \log_4(64) - \log_4(16) \), we will follow these steps:
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Apply the Quotient Rule: The Quotient Rule states that for any logarithmic expression \( \log_b(a) - \log_b(c) = \log_b\left(\frac{a}{c}\right) \).
Therefore, we can write: \[ \log_4(64) - \log_4(16) = \log_4\left(\frac{64}{16}\right) \]
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Calculate the fraction: Next, we need to simplify \(\frac{64}{16}\): \[ \frac{64}{16} = 4 \]
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Rewrite the logarithm using the simplified fraction: Now we can substitute this back into our expression: \[ \log_4(64) - \log_4(16) = \log_4(4) \]
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Evaluate \(\log_4(4)\): Now we need to evaluate \( \log_4(4) \): \[ \log_4(4) = 1 \]
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Calculate \(\log_4(64)\) and \(\log_4(16)\) separately: We can also compute these individually for verification:
- \( 64 = 4^3 \) so \( \log_4(64) = 3 \)
- \( 16 = 4^2 \) so \( \log_4(16) = 2 \)
Now substituting back gives: \[ \log_4(64) - \log_4(16) = 3 - 2 = 1 \]
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Conclusion: Since both methods yield the same result, we can conclude: \[ \log_4(64) - \log_4(16) = 1 \] and \[ \log_4\left(\frac{64}{16}\right) = \log_4(4) = 1 \]
Thus, we have verified that the Quotient Rule of Logarithms works for the base 4 expression \( \log_4(64) - \log_4(16) \).