Question
Which option best describes how to show that the Product Rule of Logarithms works for the base 4 expression
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
.
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
.
Answers
GPT-4o mini
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 \( \log_4 64 \).**
### 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 \( \log_4 64 \).**