Question

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

Answer 1

To show that the Quotient Rule of Logarithms works for the expression \( \log_4 64 - \log_4 16 \), we start by applying the Quotient Rule itself.

The Quotient Rule states that:

\[
\log_b \left( \frac{M}{N} \right) = \log_b M - \log_b N
\]

In our case, we want to verify that:

\[
\log_4 64 - \log_4 16 = \log_4 \left( \frac{64}{16} \right)
\]

First, we will calculate \( \frac{64}{16} \):

\[
\frac{64}{16} = 4
\]

Now we have:

\[
\log_4 64 - \log_4 16 = \log_4 4
\]

Next, we can evaluate \( \log_4 4 \):

Since \( 4 = 4^1 \), it follows that:

\[
\log_4 4 = 1
\]

Next, we will calculate \( \log_4 64 \) and \( \log_4 16 \) to confirm the equality.

We know that:

\[
64 = 4^3 \quad \text{(since \( 4^3 = 64 \))}
\]

Thus,

\[
\log_4 64 = 3
\]

And for \( 16 \):

\[
16 = 4^2 \quad \text{(since \( 4^2 = 16 \))}
\]

Thus,

\[
\log_4 16 = 2
\]

Now we can substitute these values back into our original expression:

\[
\log_4 64 - \log_4 16 = 3 - 2 = 1
\]

We found that \( \log_4 64 - \log_4 16 = 1 \) and previously calculated that \( \log_4 \left( \frac{64}{16} \right) = \log_4 4 = 1 \).

Since both sides are equal, we have:

\[
\log_4 64 - \log_4 16 = \log_4 \left( \frac{64}{16} \right)
\]

Thus, we have shown that the Quotient Rule of Logarithms works:

\[
\log_4 64 - \log_4 16 = \log_4 \left( \frac{64}{16} \right)
\]

This concludes the proof.