Question

Which option best describes how to show that the Product Rule of Logarithms works for the base 4 expression log416+log44
?(1 point)
Responses

Evaluate both log416
and log44
and show that their sum is equal to the value of log464
.
Evaluate both logarithm subscript 4 baseline 16 and logarithm subscript 4 baseline 4 and show that their sum is equal to the value of logarithm subscript 4 baseline 64 .

Evaluate both log416
and log44
and show that their product is equal to the value of log420
.
Evaluate both logarithm subscript 4 baseline 16 and logarithm subscript 4 baseline 4 and show that their product is equal to the value of logarithm subscript 4 baseline 20 .

Evaluate both log416
and log44
and show that their sum is equal to the value of log420
.
Evaluate both logarithm subscript 4 baseline 16 and logarithm subscript 4 baseline 4 and show that their sum is equal to the value of logarithm subscript 4 baseline 20 .

Evaluate both log416
and log44
and show that their product is equal to the value of log464
.
Evaluate both logarithm subscript 4 baseline 16 and logarithm subscript 4 baseline 4 and show that their product is equal to the value of logarithm subscript 4 baseline 64 .
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Answer 1

To demonstrate that the Product Rule of Logarithms works for the expression \( \log_4 16 + \log_4 4 \), we want to find the sum of the two logarithms and verify it against the logarithm of the product of their arguments.

According to the Product Rule, we have:

\[
\log_b (x) + \log_b (y) = \log_b (x \cdot y)
\]

In this case, \( x = 16 \) and \( y = 4 \). Therefore:

\[
\log_4 16 + \log_4 4 = \log_4 (16 \cdot 4)
\]

Calculating \( 16 \cdot 4 \) gives \( 64 \).

Now, we can express this as:

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

Now, let's evaluate both logarithms:

1. \( \log_4 16 \): Since \( 16 = 4^2 \), \( \log_4 16 = 2 \).
2. \( \log_4 4 \): Since \( 4 = 4^1 \), \( \log_4 4 = 1 \).

Adding these gives:

\[
2 + 1 = 3
\]

Now, we evaluate \( \log_4 64 \):

Since \( 64 = 4^3 \), we have \( \log_4 64 = 3 \).

Thus, we see that:

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

The correct option that describes this process is:

**Evaluate both \( \log_4 16 \) and \( \log_4 4 \) and show that their sum is equal to the value of \( \log_4 64 \).**