Question

Use the Quotient Rule of Logarithms to simplify this expression showing your steps algebraically.

log464−log416

Answer 1

To simplify the expression \( \log_4 64 - \log_4 16 \) using the Quotient Rule of Logarithms, we will follow these steps:

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

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

3. **Substitute back into the logarithm**: Now we substitute back into our logarithm:
\[
\log_4 \left( \frac{64}{16} \right) = \log_4 4
\]

4. **Evaluate the logarithm**: We know that \( \log_4 4 = 1 \) because \( 4^1 = 4 \).

Thus, the simplified expression is:
\[
\log_4 64 - \log_4 16 = 1
\]