The Quotient Rule of Logarithms states that for any logarithm \(\log_b(a) - \log_b(c)\), it can be rewritten as:
\[ \log_b\left(\frac{a}{c}\right) \]
In this case, we want to verify this for the expression \( \log_4(64) - \log_4(16) \).
First, we can apply the Quotient Rule:
\[ \log_4(64) - \log_4(16) = \log_4\left(\frac{64}{16}\right) \]
Now, let's calculate \(\frac{64}{16}\):
\[ \frac{64}{16} = 4 \]
Thus, we can rewrite the expression as:
\[ \log_4(64) - \log_4(16) = \log_4(4) \]
Now we can evaluate \(\log_4(4)\). Since \(4\) is equal to \(4^1\), we have:
\[ \log_4(4) = 1 \]
Next, we will directly compute \(\log_4(64)\) and \(\log_4(16)\) to confirm our calculation.
- Calculate \(\log_4(64)\):
Since \(64 = 4^3\), we have:
\[ \log_4(64) = \log_4(4^3) = 3 \]
- Calculate \(\log_4(16)\):
Since \(16 = 4^2\), we have:
\[ \log_4(16) = \log_4(4^2) = 2 \]
Now, we can compute \( \log_4(64) - \log_4(16) \):
\[ \log_4(64) - \log_4(16) = 3 - 2 = 1 \]
We have confirmed that both sides match:
\[ \log_4(64) - \log_4(16) = \log_4(4) = 1 \]
Thus, this shows that the Quotient Rule of Logarithms holds for the given expression:
\[ \log_4(64) - \log_4(16) = \log_4\left(\frac{64}{16}\right) \]
as expected.