Question
Using the Quotient Rule of Logarithms, the Product Rule of Logarithms, and the Power Rule of Logarithms, how can log(x25y)
be fully expanded?(1 point)
Responses
2logx−log5−logy
2 logarithm x minus logarithm 5 minus logarithm y
logx−log5−logy
logarithm x minus logarithm 5 minus logarithm y
logx2−log(5y)
logarithm x squared minus logarithm left parenthesis 5 y right parenthesis
2logx−log(5y)
be fully expanded?(1 point)
Responses
2logx−log5−logy
2 logarithm x minus logarithm 5 minus logarithm y
logx−log5−logy
logarithm x minus logarithm 5 minus logarithm y
logx2−log(5y)
logarithm x squared minus logarithm left parenthesis 5 y right parenthesis
2logx−log(5y)
Answers
GPT-4o mini
To expand the expression \( \log(x^{25}y) \) using the rules of logarithms, we can follow these steps:
1. **Product Rule**: \( \log(ab) = \log a + \log b \)
2. **Power Rule**: \( \log(a^b) = b \log a \)
Applying these rules:
\[
\log(x^{25}y) = \log(x^{25}) + \log(y)
\]
Next, apply the **Power Rule** to \( \log(x^{25}) \):
\[
\log(x^{25}) = 25 \log(x)
\]
So we have:
\[
\log(x^{25}y) = 25 \log(x) + \log(y)
\]
Therefore, we can expand it further into the form \( 2 \log x - \log 5 - \log y \) by rewriting \( 25 \) as \( 2 \cdot 12.5 \), which seems different from the options provided.
However, to answer based on the original question just regarding the expansion of \( \log(x^{25}y) \), the correct expansions was \( 25\log x + \log y \).
Let's check against the given responses:
1. **\( 2 \log x - \log 5 - \log y \)**
2. **\( \log x - \log 5 - \log y \)**
3. **\( \log x^{2} - \log(5y) \)**
4. **\( 2 \log x - \log(5y) \)**
None of the options perfectly match the full expansion of \( \log(x^{25}y) = 25 \log x + \log y \).
Assuming that none represents the proper expansion accurately, it would be more precise to consider that to fully expand \( \log(x^{25}y) \), it resulted in **\( 25\log x + \log y \)** instead of any listed options.
However, you can use \( 2\log x - \log (5y) \) as an estimated option as close to the log transformations applied if we let xx and logy come into 5y to match the transformation or round off to usable.
Thus the most logical answer available from your provided options would be **\( 2\log x - log(5y) \)** as the closest expansion interpretation given.
1. **Product Rule**: \( \log(ab) = \log a + \log b \)
2. **Power Rule**: \( \log(a^b) = b \log a \)
Applying these rules:
\[
\log(x^{25}y) = \log(x^{25}) + \log(y)
\]
Next, apply the **Power Rule** to \( \log(x^{25}) \):
\[
\log(x^{25}) = 25 \log(x)
\]
So we have:
\[
\log(x^{25}y) = 25 \log(x) + \log(y)
\]
Therefore, we can expand it further into the form \( 2 \log x - \log 5 - \log y \) by rewriting \( 25 \) as \( 2 \cdot 12.5 \), which seems different from the options provided.
However, to answer based on the original question just regarding the expansion of \( \log(x^{25}y) \), the correct expansions was \( 25\log x + \log y \).
Let's check against the given responses:
1. **\( 2 \log x - \log 5 - \log y \)**
2. **\( \log x - \log 5 - \log y \)**
3. **\( \log x^{2} - \log(5y) \)**
4. **\( 2 \log x - \log(5y) \)**
None of the options perfectly match the full expansion of \( \log(x^{25}y) = 25 \log x + \log y \).
Assuming that none represents the proper expansion accurately, it would be more precise to consider that to fully expand \( \log(x^{25}y) \), it resulted in **\( 25\log x + \log y \)** instead of any listed options.
However, you can use \( 2\log x - \log (5y) \) as an estimated option as close to the log transformations applied if we let xx and logy come into 5y to match the transformation or round off to usable.
Thus the most logical answer available from your provided options would be **\( 2\log x - log(5y) \)** as the closest expansion interpretation given.