Question

Show that the Quotient Rule of Logarithms works for the base 4 expression log_4 64−log_4 16.

Answer 1

To show that the Quotient Rule of Logarithms works for the base 4 expression \( \log_4 64 - \log_4 16 \), we can use the Quotient Rule of Logarithms, which states:

\[
\log_b \frac{M}{N} = \log_b M - \log_b N
\]

In this case, we can set \( M = 64 \) and \( N = 16 \). Thus, we can rewrite the expression using the Quotient Rule:

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

Now, we need to compute the value of \( \frac{64}{16} \):

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

Substituting this back into our expression, we have:

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

Next, we know that:

\[
\log_4 4 = 1
\]

Now, we can calculate \( \log_4 64 \) and \( \log_4 16 \) separately to verify our results:

1. Calculating \( \log_4 64 \):
- Since \( 64 = 4^3 \),
\[
\log_4 64 = 3
\]

2. Calculating \( \log_4 16 \):
- Since \( 16 = 4^2 \),
\[
\log_4 16 = 2
\]

Now substituting these values back into our original expression, we get:

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

Thus, we have shown that:

\[
\log_4 64 - \log_4 16 = \log_4 4 = 1
\]

Hence, the Quotient Rule of Logarithms works for the base 4 expression \( \log_4 64 - \log_4 16 \).