To find the expression equivalent to [tex]\(-2 \log_2 x + 4 \log_2 y + 4 \log_2 z\)[/tex], we should use the properties of logarithms. Let's go through the problem step-by-step:
1. Logarithm Power Rule:
The power rule states that [tex]\( a \log_b c = \log_b(c^a) \)[/tex]. We apply this rule separately to each term in the given expression:
[tex]\[
-2 \log_2 x = \log_2(x^{-2})
\][/tex]
[tex]\[
4 \log_2 y = \log_2(y^4)
\][/tex]
[tex]\[
4 \log_2 z = \log_2(z^4)
\][/tex]
2. Combining Logarithms:
Now, we combine these logarithms using the logarithm addition and subtraction rules. The addition rule is [tex]\(\log_b(a) + \log_b(b) = \log_b(a \cdot b)\)[/tex] and the subtraction rule is [tex]\(\log_b(a) - \log_b(b) = \log_b\left(\frac{a}{b}\right)\)[/tex]. Combining the terms:
[tex]\[
\log_2(x^{-2}) + \log_2(y^4) + \log_2(z^4)
\][/tex]
Combining all these logarithms into one, according to the rules:
[tex]\[
\log_2 \left( x^{-2} \cdot y^4 \cdot z^4 \right)
\][/tex]
3. Simplify the Expression:
Simplify the inside of the logarithm:
[tex]\[
x^{-2} \cdot y^4 \cdot z^4 = \frac{y^4 \cdot z^4}{x^2}
\][/tex]
Therefore:
[tex]\[
\log_2 \left( \frac{y^4 z^4}{x^2} \right)
\][/tex]
So, the equivalent expression to [tex]\(-2 \log_2 x + 4 \log_2 y + 4 \log_2 z\)[/tex] is:
[tex]\[
\log_2 \left( \frac{y^4 z^4}{x^2} \right)
\][/tex]
Thus, the correct option is:
[tex]\[
\boxed{\log_2\left(\frac{y^4 z^4}{x^2}\right)}
\][/tex]
