Let's analyze the given problem step-by-step:
We want to determine which expressions are equivalent to [tex]\(\log_4\left(\frac{1}{4} x^2\right)\)[/tex].
### Original Expression:
[tex]\(\log_4\left(\frac{1}{4} x^2\right)\)[/tex]
#### Expression (1):
[tex]\[
-2 + 2 \log_4 x
\][/tex]
We will use logarithm properties to compare:
1. Let’s break down [tex]\(\log_4\left(\frac{1}{4} x^2\right)\)[/tex] using properties of log.
[tex]\[
\log_4\left(\frac{1}{4} x^2\right) = \log_4\left(\frac{1}{4}\right) + \log_4(x^2)
\][/tex]
2. Simplify each component:
[tex]\[
\log_4\left(\frac{1}{4}\right) = \log_4\left(4^{-1}\right) = -1
\][/tex]
[tex]\[
\log_4(x^2) = 2 \log_4(x)
\][/tex]
So,
[tex]\[
\log_4\left(\frac{1}{4} x^2\right) = -1 + 2 \log_4(x)
\][/tex]
Thus, [tex]\(\log_4\left(\frac{1}{4} x^2\right) = -1 + 2 \log_4(x)\)[/tex].
Expression (1): [tex]\(-2 + 2 \log_4 x\)[/tex] is not equivalent to this because it results in [tex]\(-2 + 2 \log_4 x\)[/tex].
### Expression (2):
[tex]\[
2 \log_4\left(\frac{1}{4}\right) - \log_4 x^2
\][/tex]
Using logarithm properties:
[tex]\[
2 \log_4\left(\frac{1}{4}\right) = 2(-1) = -2
\][/tex]
[tex]\[
\log_4(x^2) = 2 \log_4(x)
\][/tex]
Then,
[tex]\[
2 \log_4\left(\frac{1}{4}\right) - \log_4(x^2) = -2 - 2 \log_4(x)
\][/tex]
### Expression (3):
[tex]\[
2 \log_4\left(\frac{1}{4} x\right)
\][/tex]
Let's simplify this expression:
[tex]\[
\log_4\left(\frac{1}{4} x\right) \text{ can be split as } \log_4\left(\frac{1}{4}\right) + \log_4(x)
\][/tex]
[tex]\[
\log_4\left(\frac{1}{4}\right) = -1
\][/tex]
[tex]\[
2 \log_4\left(\frac{1}{4} x\right) = 2(-1 + \log_4(x)) = -2 + 2 \log_4(x)
\][/tex]
### Expression (4):
[tex]\[
-1 + 2 \log_4 x
\][/tex]
As discussed earlier,
[tex]\(\log_4\left(\frac{1}{4} x^2\right) = -1 + 2 \log_4(x)\)[/tex]
Thus, this expression is equivalent.
### Expression (5):
[tex]\[
\log_4\left(\frac{1}{4}\right) + \log_4 x^2
\][/tex]
Using properties of logarithms:
[tex]\[
\log_4\left(\frac{1}{4}\right) = -1,\quad \text{and} \quad \log_4(x^2) = 2 \log_4(x)
\][/tex]
Combining these:
[tex]\(-1 + 2 \log_4(x)\)[/tex]
Thus, this expression is equivalent.
### Summary:
From the analysis, the expressions equivalent to [tex]\(\log_4\left(\frac{1}{4} x^2\right)\)[/tex] are:
- [tex]\(\boxed{-1 + 2 \log_4 x}\)[/tex]
- [tex]\(\boxed{\log_4\left(\frac{1}{4}\right) + \log_4 x^2}\)[/tex]
So the correct answers are:
- [tex]\(-1 + 2 \log _4 x\)[/tex]
- [tex]\(\log _4\left(\frac{1}{4}\right) + \log _4 x^2\)[/tex]
