Question

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

log
4
64
−
log
4
16

Answer 1

To simplify the expression \(\log_4 64 - \log_4 16\) using the Quotient Rule of Logarithms, we start by recalling the Quotient Rule itself. The Quotient Rule states that:

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

Using this rule, we can combine the two logarithmic terms. Here, \(M\) is \(64\) and \(N\) is \(16\):

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

Now, we need to simplify \(\frac{64}{16}\):

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

Thus, we can rewrite the expression as:

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

Now, we know that \(\log_b b = 1\), so:

\[
\log_4 4 = 1
\]

Therefore, the final simplified expression is:

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

In conclusion, the simplification of \(\log_4 64 - \log_4 16\) results in \(1\).