Which option best describes how to show that the Product Rule of Logarithms works for the base 4 expression

Question

log
4
16
+
log
4
4
?(1 point)
Responses

Evaluate both
log
4
16
and
log
4
4
and show that their sum is equal to the value of
log
4
20
.
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
log
4
16
and
log
4
4
and show that their product is equal to the value of
log
4
20
.
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
log
4
16
and
log
4
4
and show that their product is equal to the value of
log
4
64
.
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 .

Evaluate both
log
4
16
and
log
4
4
and show that their sum is equal to the value of
log
4
64
.

Answer 1

To show that the Product Rule of Logarithms works for the base 4 expression \( \log_4 16 + \log_4 4 \), we can evaluate both logarithms and check their sum against the logarithm of the product of the arguments.

Calculation:

1. First, evaluate \( \log_4 16 \): \[ \log_4 16 = \log_4 (4^2) = 2 \]

2. Next, evaluate \( \log_4 4 \): \[ \log_4 4 = 1 \]

3. Now, add the two results: \[ \log_4 16 + \log_4 4 = 2 + 1 = 3 \]

4. Finally, check if this sum equals \( \log_4 (16 \cdot 4) = \log_4 64 \): \[ 16 \cdot 4 = 64 \] \[ \log_4 64 = \log_4 (4^3) = 3 \]

Conclusion:

Thus, we have shown that \( \log_4 16 + \log_4 4 = \log_4 64 \), which validates the Product Rule for logarithms in this case.

Correct Option:

The response that fits this explanation is: Evaluate both \( \log_4 16 \) and \( \log_4 4 \) and show that their sum is equal to the value of \( \log_4 64 \).