To determine which of the given expressions are equivalent to [tex]\(\log_4\left(\frac{1}{4} x^2\right)\)[/tex], we need to simplify [tex]\(\log_4\left(\frac{1}{4} x^2\right)\)[/tex] step-by-step using logarithm properties.
First, recall the logarithm properties that we need:
1. [tex]\(\log_b(mn) = \log_b(m) + \log_b(n)\)[/tex]
2. [tex]\(\log_b\left(\frac{m}{n}\right) = \log_b(m) - \log_b(n)\)[/tex]
3. [tex]\(\log_b(m^n) = n \cdot \log_b(m)\)[/tex]
Let's apply these to [tex]\(\log_4\left(\frac{1}{4} x^2\right)\)[/tex]:
### Step 1: Split the logarithm using property 2
[tex]\[
\log_4\left(\frac{1}{4} x^2\right) = \log_4\left(\frac{1}{4}\right) + \log_4(x^2)
\][/tex]
### Step 2: Simplify [tex]\(\log_4\left(\frac{1}{4}\right)\)[/tex]
Note that [tex]\(\frac{1}{4} = 4^{-1}\)[/tex], so:
[tex]\[
\log_4\left(\frac{1}{4}\right) = \log_4\left(4^{-1}\right) = -1
\][/tex]
### Step 3: Simplify [tex]\(\log_4(x^2)\)[/tex]
Using property 3:
[tex]\[
\log_4(x^2) = 2 \cdot \log_4(x)
\][/tex]
### Step 4: Substitute back
Combine the results from Steps 2 and 3:
[tex]\[
\log_4\left(\frac{1}{4} x^2\right) = -1 + 2 \cdot \log_4(x)
\][/tex]
Now, let's evaluate each given expression to see if it matches [tex]\(\log_4\left(\frac{1}{4} x^2\right) = -1 + 2 \cdot \log_4(x)\)[/tex]:
1. [tex]\(2 \log_4\left(\frac{1}{4}\right) - \log_4(x^2)\)[/tex]
- Simplify: [tex]\(2 \cdot (-1) - 2 \log_4(x) = -2 - 2 \log_4(x)\)[/tex]
- This does not match the simplified expression.
2. [tex]\(\log_4\left(\frac{1}{4}\right) + \log_4(x^2)\)[/tex]
- Simplify: [tex]\(-1 + 2 \log_4(x)\)[/tex]
- This matches the simplified expression.
3. [tex]\(2 \log_4\left(\frac{1}{4} x\right)\)[/tex]
- Simplify: [tex]\(2 \left(\log_4\left(\frac{1}{4}\right) + \log_4(x)\right) = 2 \left(-1 + \log_4(x)\right) = -2 + 2 \log_4(x)\)[/tex]
- This matches the simplified expression.
4. [tex]\(-1 + 2 \log_4(x)\)[/tex]
- Simplify: [tex]\(-1 + 2 \log_4(x)\)[/tex]
- This matches the simplified expression.
5. [tex]\(-2 + 2 \log_4(x)\)[/tex]
- This does not match the simplified expression.
Based on the simplifications and comparisons, the expressions equivalent to [tex]\(\log_4\left(\frac{1}{4} x^2\right)\)[/tex] are:
[tex]\[
\boxed{\log_4\left(\frac{1}{4}\right) + \log_4(x^2), \ 2 \log_4\left(\frac{1}{4} x\right), \ -1 + 2 \log_4(x)}
\][/tex]
