To determine which expression is equivalent to [tex]\(-2 \log_2 x + 4 \log_2 y + 4 \log_2 z\)[/tex], let's follow a detailed step-by-step simplification process.
### Step 1: Apply the Power Rule of Logarithms
The power rule of logarithms states that [tex]\(a \log_b(c) = \log_b(c^a)\)[/tex]. We'll apply this rule to each term separately:
1. [tex]\(-2 \log_2 x\)[/tex]:
[tex]\[
-2 \log_2 x = \log_2(x^{-2})
\][/tex]
2. [tex]\(4 \log_2 y\)[/tex]:
[tex]\[
4 \log_2 y = \log_2(y^4)
\][/tex]
3. [tex]\(4 \log_2 z\)[/tex]:
[tex]\[
4 \log_2 z = \log_2(z^4)
\][/tex]
After applying the power rule to each term, the expression becomes:
[tex]\[
\log_2(x^{-2}) + \log_2(y^4) + \log_2(z^4)
\][/tex]
### Step 2: Apply the Addition Rule of Logarithms
The addition rule of logarithms states that [tex]\( \log_b(A) + \log_b(B) = \log_b(AB) \)[/tex]. We will combine the logarithms using this rule:
First, combine [tex]\(\log_2(x^{-2})\)[/tex] and [tex]\(\log_2(y^4)\)[/tex]:
[tex]\[
\log_2(x^{-2}) + \log_2(y^4) = \log_2(x^{-2} y^4)
\][/tex]
Next, combine this result with [tex]\(\log_2(z^4)\)[/tex]:
[tex]\[
\log_2(x^{-2} y^4) + \log_2(z^4) = \log_2(x^{-2} y^4 z^4)
\][/tex]
### Step 3: Simplify the Expression Inside the Logarithm
We combine the terms inside the logarithm:
[tex]\[
\log_2(x^{-2} y^4 z^4)
\][/tex]
Recognize that [tex]\(x^{-2} = \frac{1}{x^2}\)[/tex]. Therefore, the expression inside the logarithm can be rewritten as:
[tex]\[
x^{-2} y^4 z^4 = \frac{y^4 z^4}{x^2}
\][/tex]
Thus, the entire expression is:
[tex]\[
\log_2\left(\frac{y^4 z^4}{x^2}\right)
\][/tex]
Therefore, the expression that is equivalent to [tex]\(-2 \log_2 x + 4 \log_2 y + 4 \log_2 z\)[/tex] is:
[tex]\[
\boxed{\log_2\left(\frac{y^4 z^4}{x^2}\right)}
\][/tex]
