Question
Expand log subscript start fraction 1 over 2 end fraction baseline left parenthesis start fraction 3x superscript 2 baseline over 2 end fraction right parenthesis using the properties and rules for logarithms.
(1 point)
Responses
log subscript start fraction 1 over 2 end fraction baseline left parenthesis 3 right parenthesis plus 2 log left parenthesis x right parenthesis minus log subscript start fraction 1 over 2 end fraction baseline left parenthesis 2 right parenthesis
Image with alt text: log subscript start fraction 1 over 2 end fraction baseline left parenthesis 3 right parenthesis plus 2 log left parenthesis x right parenthesis minus log subscript start fraction 1 over 2 end fraction baseline left parenthesis 2 right parenthesis
2 log subscript start fraction 1 over 2 end fraction baseline left parenthesis 3 right parenthesis plus 2 log subscript start fraction 1 over 2 end fraction baseline left parenthesis x right parenthesis plus 1
Image with alt text: 2 log subscript start fraction 1 over 2 end fraction baseline left parenthesis 3 right parenthesis plus 2 log subscript start fraction 1 over 2 end fraction baseline left parenthesis x right parenthesis plus 1
2 log subscript start fraction 1 over 2 end fraction baseline left parenthesis 3x right parenthesis plus 1
Image with alt text: 2 log subscript start fraction 1 over 2 end fraction baseline left parenthesis 3x right parenthesis plus 1
2 log subscript start fraction 1 over 2 end fraction baseline left parenthesis 3x right parenthesis minus log subscript start fraction 1 over 2 end fraction baseline left parenthesis 2 right parenthesis
(1 point)
Responses
log subscript start fraction 1 over 2 end fraction baseline left parenthesis 3 right parenthesis plus 2 log left parenthesis x right parenthesis minus log subscript start fraction 1 over 2 end fraction baseline left parenthesis 2 right parenthesis
Image with alt text: log subscript start fraction 1 over 2 end fraction baseline left parenthesis 3 right parenthesis plus 2 log left parenthesis x right parenthesis minus log subscript start fraction 1 over 2 end fraction baseline left parenthesis 2 right parenthesis
2 log subscript start fraction 1 over 2 end fraction baseline left parenthesis 3 right parenthesis plus 2 log subscript start fraction 1 over 2 end fraction baseline left parenthesis x right parenthesis plus 1
Image with alt text: 2 log subscript start fraction 1 over 2 end fraction baseline left parenthesis 3 right parenthesis plus 2 log subscript start fraction 1 over 2 end fraction baseline left parenthesis x right parenthesis plus 1
2 log subscript start fraction 1 over 2 end fraction baseline left parenthesis 3x right parenthesis plus 1
Image with alt text: 2 log subscript start fraction 1 over 2 end fraction baseline left parenthesis 3x right parenthesis plus 1
2 log subscript start fraction 1 over 2 end fraction baseline left parenthesis 3x right parenthesis minus log subscript start fraction 1 over 2 end fraction baseline left parenthesis 2 right parenthesis
Answers
GPT-4o mini
To expand the expression \(\log_{\frac{1}{2}}\left(\frac{3x^2}{2}\right)\) using the properties and rules of logarithms, we can apply the following logarithmic properties:
1. \(\log_b\left(\frac{M}{N}\right) = \log_b(M) - \log_b(N)\) (the quotient rule)
2. \(\log_b(M^k) = k \cdot \log_b(M)\) (the power rule)
Now, let's use these properties to expand the given logarithm:
1. Start with the original expression:
\[
\log_{\frac{1}{2}}\left(\frac{3x^2}{2}\right)
\]
2. Apply the quotient rule:
\[
= \log_{\frac{1}{2}}(3x^2) - \log_{\frac{1}{2}}(2)
\]
3. Now, we can expand \(\log_{\frac{1}{2}}(3x^2)\) using the product rule (\(\log_b(MN) = \log_b(M) + \log_b(N)\)):
\[
= \log_{\frac{1}{2}}(3) + \log_{\frac{1}{2}}(x^2) - \log_{\frac{1}{2}}(2)
\]
4. Now, apply the power rule to \(\log_{\frac{1}{2}}(x^2)\):
\[
= \log_{\frac{1}{2}}(3) + 2\log_{\frac{1}{2}}(x) - \log_{\frac{1}{2}}(2)
\]
Putting it all together, we have:
\[
\log_{\frac{1}{2}}(3) + 2\log_{\frac{1}{2}}(x) - \log_{\frac{1}{2}}(2)
\]
So the expanded logarithmic expression is:
\[
\log_{\frac{1}{2}}(3) + 2 \log_{\frac{1}{2}}(x) - \log_{\frac{1}{2}}(2)
\]
Looking at the provided responses, the equivalent response is:
\[
\log_{\frac{1}{2}}(3) + 2 \log(x) - \log_{\frac{1}{2}}(2)
\]
Thus, that corresponds to the first option you presented.
1. \(\log_b\left(\frac{M}{N}\right) = \log_b(M) - \log_b(N)\) (the quotient rule)
2. \(\log_b(M^k) = k \cdot \log_b(M)\) (the power rule)
Now, let's use these properties to expand the given logarithm:
1. Start with the original expression:
\[
\log_{\frac{1}{2}}\left(\frac{3x^2}{2}\right)
\]
2. Apply the quotient rule:
\[
= \log_{\frac{1}{2}}(3x^2) - \log_{\frac{1}{2}}(2)
\]
3. Now, we can expand \(\log_{\frac{1}{2}}(3x^2)\) using the product rule (\(\log_b(MN) = \log_b(M) + \log_b(N)\)):
\[
= \log_{\frac{1}{2}}(3) + \log_{\frac{1}{2}}(x^2) - \log_{\frac{1}{2}}(2)
\]
4. Now, apply the power rule to \(\log_{\frac{1}{2}}(x^2)\):
\[
= \log_{\frac{1}{2}}(3) + 2\log_{\frac{1}{2}}(x) - \log_{\frac{1}{2}}(2)
\]
Putting it all together, we have:
\[
\log_{\frac{1}{2}}(3) + 2\log_{\frac{1}{2}}(x) - \log_{\frac{1}{2}}(2)
\]
So the expanded logarithmic expression is:
\[
\log_{\frac{1}{2}}(3) + 2 \log_{\frac{1}{2}}(x) - \log_{\frac{1}{2}}(2)
\]
Looking at the provided responses, the equivalent response is:
\[
\log_{\frac{1}{2}}(3) + 2 \log(x) - \log_{\frac{1}{2}}(2)
\]
Thus, that corresponds to the first option you presented.