Which expressions are equivalent to [tex]\(\log _4\left(\frac{1}{4} x^2\right)\)[/tex]?

A. [tex]\(-2 + 2 \log _4 x\)[/tex]

B. [tex]\(2 \log _4\left(\frac{1}{4}\right) - \log _4 x^2\)[/tex]

C. [tex]\(2 \log _4\left(\frac{1}{4} x\right)\)[/tex]

D. [tex]\(-1 + 2 \log _4 x\)[/tex]

E. [tex]\(\log _4\left(\frac{1}{4}\right) + \log _4 x^2\)[/tex]



Answer :

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]