Question

Show that the Quotient Rule of Logarithms works for the base 4 expression log464−log416
.(4 points)

Answer 1

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:

1. **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:
\[
\log_4(64) - \log_4(16) = \log_4\left(\frac{64}{16}\right)
\]

2. **Calculate the fraction:**
Next, we need to simplify \(\frac{64}{16}\):
\[
\frac{64}{16} = 4
\]

3. **Rewrite the logarithm using the simplified fraction:**
Now we can substitute this back into our expression:
\[
\log_4(64) - \log_4(16) = \log_4(4)
\]

4. **Evaluate \(\log_4(4)\):**
Now we need to evaluate \( \log_4(4) \):
\[
\log_4(4) = 1
\]

5. **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:
\[
\log_4(64) - \log_4(16) = 3 - 2 = 1
\]

6. **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 \( \log_4(64) - \log_4(16) \).