Question

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

Answer 1

The Quotient Rule of Logarithms states that for any logarithm \(\log_b(a) - \log_b(c)\), it can be rewritten as:

\[
\log_b\left(\frac{a}{c}\right)
\]

In this case, we want to verify this for the expression \( \log_4(64) - \log_4(16) \).

First, we can apply the Quotient Rule:

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

Now, let's calculate \(\frac{64}{16}\):

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

Thus, we can rewrite the expression as:

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

Now we can evaluate \(\log_4(4)\). Since \(4\) is equal to \(4^1\), we have:

\[
\log_4(4) = 1
\]

Next, we will directly compute \(\log_4(64)\) and \(\log_4(16)\) to confirm our calculation.

1. Calculate \(\log_4(64)\):

Since \(64 = 4^3\), we have:

\[
\log_4(64) = \log_4(4^3) = 3
\]

2. Calculate \(\log_4(16)\):

Since \(16 = 4^2\), we have:

\[
\log_4(16) = \log_4(4^2) = 2
\]

Now, we can compute \( \log_4(64) - \log_4(16) \):

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

We have confirmed that both sides match:

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

Thus, this shows that the Quotient Rule of Logarithms holds for the given expression:

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

as expected.