Question
A student was asked to expand and simplify log_2 (8x/y)
completely. The student's work is shown below. log_2 (8x.y) - log_2 y = log_2 + log_2 x - log_2 y
Answers
GPT-4o mini
To expand and simplify \( \log_2 \left( \frac{8x}{y} \right) \), we can use the properties of logarithms.
1. **Start with the expression**:
\[
\log_2 \left( \frac{8x}{y} \right)
\]
2. **Use the quotient rule of logarithms**:
\[
\log_2 \left( \frac{8x}{y} \right) = \log_2(8x) - \log_2(y)
\]
3. **Now apply the product rule of logarithms to \( \log_2(8x) \)**:
\[
\log_2(8x) = \log_2(8) + \log_2(x)
\]
4. **Next, calculate \( \log_2(8) \)**:
Since \( 8 = 2^3 \), we have \( \log_2(8) = 3 \).
5. **Substituting this back, we have**:
\[
\log_2(8x) = 3 + \log_2(x)
\]
6. **Combine this into the earlier expression**:
\[
\log_2 \left( \frac{8x}{y} \right) = (3 + \log_2(x)) - \log_2(y)
\]
7. **So the final expression is**:
\[
\log_2 \left( \frac{8x}{y} \right) = 3 + \log_2(x) - \log_2(y)
\]
Thus, the correct expansion and simplification of \( \log_2 \left( \frac{8x}{y} \right) \) is:
\[
3 + \log_2(x) - \log_2(y)
\]
Please check if the student's work accurately reflects these steps, but from the process, the student's final expression should be \( 3 + \log_2(x) - \log_2(y) \) rather than the introduced \( log_2 + \log_2 x - log_2 y \) in their work.
1. **Start with the expression**:
\[
\log_2 \left( \frac{8x}{y} \right)
\]
2. **Use the quotient rule of logarithms**:
\[
\log_2 \left( \frac{8x}{y} \right) = \log_2(8x) - \log_2(y)
\]
3. **Now apply the product rule of logarithms to \( \log_2(8x) \)**:
\[
\log_2(8x) = \log_2(8) + \log_2(x)
\]
4. **Next, calculate \( \log_2(8) \)**:
Since \( 8 = 2^3 \), we have \( \log_2(8) = 3 \).
5. **Substituting this back, we have**:
\[
\log_2(8x) = 3 + \log_2(x)
\]
6. **Combine this into the earlier expression**:
\[
\log_2 \left( \frac{8x}{y} \right) = (3 + \log_2(x)) - \log_2(y)
\]
7. **So the final expression is**:
\[
\log_2 \left( \frac{8x}{y} \right) = 3 + \log_2(x) - \log_2(y)
\]
Thus, the correct expansion and simplification of \( \log_2 \left( \frac{8x}{y} \right) \) is:
\[
3 + \log_2(x) - \log_2(y)
\]
Please check if the student's work accurately reflects these steps, but from the process, the student's final expression should be \( 3 + \log_2(x) - \log_2(y) \) rather than the introduced \( log_2 + \log_2 x - log_2 y \) in their work.