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
Did the student simplify the expression completely? Explain your reasoning.
completely. The student's work is shown below. log_2 (8x.y) - log_2 y = log_2 + log_2 x - log_2 y
Did the student simplify the expression completely? Explain your reasoning.
Answers
GPT-4o mini
Let's analyze the student's work step by step and figure out if they simplified the expression completely.
The given expression is:
\[
\log_2\left(\frac{8x}{y}\right)
\]
Using the properties of logarithms, specifically the quotient rule, we can split this into two parts:
\[
\log_2(8x) - \log_2(y)
\]
Next, we can apply the product rule for logarithms to further simplify the first term:
\[
\log_2(8) + \log_2(x) - \log_2(y)
\]
Now, we know that \(8 = 2^3\), so:
\[
\log_2(8) = 3
\]
Therefore, we can substitute this into our expression:
\[
3 + \log_2(x) - \log_2(y)
\]
So the complete simplification results in:
\[
3 + \log_2(x) - \log_2(y)
\]
Now, let's review the student's work:
The student wrote:
\[
\log_2 (8x) - \log_2(y) = \log_2 + \log_2(x) - \log_2(y)
\]
This part contains a mistake; they incorrectly stated \(\log_2(8x)\) as \(\log_2 + \log_2(x)\), which is not correct. The correct form would include \(\log_2(8)\), which should have been shown as a value of \(3\) instead of just writing \(\log_2\).
In conclusion, the student did not simplify the expression completely. The correct final result should include \(3\) from \(\log_2(8)\). Thus, the final simplified expression should be:
\[
3 + \log_2(x) - \log_2(y)
\]
The given expression is:
\[
\log_2\left(\frac{8x}{y}\right)
\]
Using the properties of logarithms, specifically the quotient rule, we can split this into two parts:
\[
\log_2(8x) - \log_2(y)
\]
Next, we can apply the product rule for logarithms to further simplify the first term:
\[
\log_2(8) + \log_2(x) - \log_2(y)
\]
Now, we know that \(8 = 2^3\), so:
\[
\log_2(8) = 3
\]
Therefore, we can substitute this into our expression:
\[
3 + \log_2(x) - \log_2(y)
\]
So the complete simplification results in:
\[
3 + \log_2(x) - \log_2(y)
\]
Now, let's review the student's work:
The student wrote:
\[
\log_2 (8x) - \log_2(y) = \log_2 + \log_2(x) - \log_2(y)
\]
This part contains a mistake; they incorrectly stated \(\log_2(8x)\) as \(\log_2 + \log_2(x)\), which is not correct. The correct form would include \(\log_2(8)\), which should have been shown as a value of \(3\) instead of just writing \(\log_2\).
In conclusion, the student did not simplify the expression completely. The correct final result should include \(3\) from \(\log_2(8)\). Thus, the final simplified expression should be:
\[
3 + \log_2(x) - \log_2(y)
\]