Asked by Jaz
Write the expression as a single logarithm.
15 left-bracket one-third log left-parenthesis x plus y right-parenthesis minus Start Fraction 1 over 5 End Fraction log left-parenthesis x minus y right-parenthesis right-bracket
(1 point)
Responses
log Start Fraction left-parenthesis x plus y right-parenthesis cubed over left-parenthesis x minus y right-parenthesis superscript 5 baseline End Fraction
Image with alt text: log Start Fraction left-parenthesis x plus y right-parenthesis cubed over left-parenthesis x minus y right-parenthesis superscript 5 baseline End Fraction
log Start Fraction left-parenthesis x plus y right-parenthesis superscript 5 baseline over left-parenthesis x minus y right-parenthesis cubed End Fraction
Image with alt text: log Start Fraction left-parenthesis x plus y right-parenthesis superscript 5 baseline over left-parenthesis x minus y right-parenthesis cubed End Fraction
negative log left-parenthesis x plus y right-parenthesis squared
Image with alt text: negative log left-parenthesis x plus y right-parenthesis squared
Start Fraction negative 1 over log left-parenthesis x plus y right-parenthesis squared End Fraction
Image with alt text: Start Fraction negative 1 over log left-parenthesis x plus y right-parenthesis squared End Fraction
15 left-bracket one-third log left-parenthesis x plus y right-parenthesis minus Start Fraction 1 over 5 End Fraction log left-parenthesis x minus y right-parenthesis right-bracket
(1 point)
Responses
log Start Fraction left-parenthesis x plus y right-parenthesis cubed over left-parenthesis x minus y right-parenthesis superscript 5 baseline End Fraction
Image with alt text: log Start Fraction left-parenthesis x plus y right-parenthesis cubed over left-parenthesis x minus y right-parenthesis superscript 5 baseline End Fraction
log Start Fraction left-parenthesis x plus y right-parenthesis superscript 5 baseline over left-parenthesis x minus y right-parenthesis cubed End Fraction
Image with alt text: log Start Fraction left-parenthesis x plus y right-parenthesis superscript 5 baseline over left-parenthesis x minus y right-parenthesis cubed End Fraction
negative log left-parenthesis x plus y right-parenthesis squared
Image with alt text: negative log left-parenthesis x plus y right-parenthesis squared
Start Fraction negative 1 over log left-parenthesis x plus y right-parenthesis squared End Fraction
Image with alt text: Start Fraction negative 1 over log left-parenthesis x plus y right-parenthesis squared End Fraction
Answers
Answered by
GPT-4o mini
To rewrite the expression \( 15 \left( \frac{1}{3} \log (x+y) - \frac{1}{5} \log (x-y) \right) \) as a single logarithm, we will first distribute the 15 across the terms:
1. Rewrite the expression:
\[
15 \left( \frac{1}{3} \log (x+y) - \frac{1}{5} \log (x-y) \right) = 15 \cdot \frac{1}{3} \log (x+y) - 15 \cdot \frac{1}{5} \log (x-y)
\]
2. Simplify the coefficients:
\[
= 5 \log (x+y) - 3 \log (x-y)
\]
3. Use the power rule of logarithms which states that \( a \log b = \log (b^a) \):
\[
= \log ((x+y)^5) - \log ((x-y)^3)
\]
4. Now, use the property of logarithms that states \( \log a - \log b = \log \left( \frac{a}{b} \right) \):
\[
= \log \left( \frac{(x+y)^5}{(x-y)^3} \right)
\]
Thus, the expression as a single logarithm is:
\[
\log \left( \frac{(x+y)^5}{(x-y)^3} \right)
\]
From the options provided, the correct response would be:
\[
\log \left( \frac{(x+y)^5}{(x-y)^3} \right)
\]
1. Rewrite the expression:
\[
15 \left( \frac{1}{3} \log (x+y) - \frac{1}{5} \log (x-y) \right) = 15 \cdot \frac{1}{3} \log (x+y) - 15 \cdot \frac{1}{5} \log (x-y)
\]
2. Simplify the coefficients:
\[
= 5 \log (x+y) - 3 \log (x-y)
\]
3. Use the power rule of logarithms which states that \( a \log b = \log (b^a) \):
\[
= \log ((x+y)^5) - \log ((x-y)^3)
\]
4. Now, use the property of logarithms that states \( \log a - \log b = \log \left( \frac{a}{b} \right) \):
\[
= \log \left( \frac{(x+y)^5}{(x-y)^3} \right)
\]
Thus, the expression as a single logarithm is:
\[
\log \left( \frac{(x+y)^5}{(x-y)^3} \right)
\]
From the options provided, the correct response would be:
\[
\log \left( \frac{(x+y)^5}{(x-y)^3} \right)
\]