Question

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

Answer 1

The Quotient Rule for logarithms states that for any logarithm with base \( b \), the following holds:

\[
\log_b \frac{a}{c} = \log_b a - \log_b c
\]

We need to demonstrate that this rule applies to the expression \( \log_4 64 - \log_4 16 \).

First, we can individually calculate \( \log_4 64 \) and \( \log_4 16 \):

1. **Calculating \( \log_4 64 \)**:
We know that \( 64 = 4^3 \), since \( 4^3 = 64 \). Therefore,
\[
\log_4 64 = \log_4 (4^3) = 3.
\]

2. **Calculating \( \log_4 16 \)**:
Similarly, \( 16 = 4^2 \), so:
\[
\log_4 16 = \log_4 (4^2) = 2.
\]

Now we can evaluate \( \log_4 64 - \log_4 16 \):
\[
log_4 64 - log_4 16 = 3 - 2 = 1.
\]

Next, let's use the Quotient Rule to see if it yields the same result. We can rewrite our original expression \( \log_4 64 - \log_4 16 \) as:
\[
\log_4 \frac{64}{16}.
\]

Now, we need to simplify \( \frac{64}{16} \):
\[
\frac{64}{16} = 4.
\]

Now we can calculate \( \log_4 4 \):
Since \( 4 = 4^1 \), we have:
\[
\log_4 4 = 1.
\]

Both methods yield the same result:
\[
log_4 64 - log_4 16 = 1,
\]
and
\[
log_4 \frac{64}{16} = 1.
\]

Thus, we have shown that the Quotient Rule of Logarithms holds for this base 4 expression, confirming that:

\[
\log_4 64 - \log_4 16 = \log_4 \frac{64}{16}.
\]