Question

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

Answer 1

To demonstrate that the Quotient Rule of Logarithms works for the expression \( \log_4 64 - \log_4 16 \), we start by applying the Quotient Rule. The Quotient Rule states that:

\[
\log_b \frac{m}{n} = \log_b m - \log_b n
\]

This means we can rewrite the expression:

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

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

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

Now substituting this back into our logarithm expression gives us:

\[
\log_4 \frac{64}{16} = \log_4 4
\]

Now, we find \( \log_4 4 \). Since \( 4 \) is the base of the logarithm, we have:

\[
\log_4 4 = 1
\]

Now, we will separately evaluate \( \log_4 64 \) and \( \log_4 16 \) to verify our calculations:

1. To find \( \log_4 64 \), express \( 64 \) as a power of \( 4 \):
\[
64 = 4^3 \quad \text{(since \( 4^3 = 64 \))}
\]

Therefore,
\[
\log_4 64 = 3
\]

2. To find \( \log_4 16 \), express \( 16 \) as a power of \( 4 \):
\[
16 = 4^2 \quad \text{(since \( 4^2 = 16 \))}
\]

Therefore,
\[
\log_4 16 = 2
\]

Now we can substitute these values back into our original expression \( \log_4 64 - \log_4 16 \) to confirm:

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

Since we derived earlier that \( \log_4 \frac{64}{16} = 1 \), we conclude that both methods yield the same result.

Thus, we have shown that:

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

and verified it. The Quotient Rule of Logarithms holds true for the given expression.