To determine which expressions are equivalent to [tex]\(\log _4\left(\frac{1}{d} x^2\right)\)[/tex], we will use properties of logarithms to simplify and compare each option to the given expression.
First, let's simplify [tex]\(\log _4\left(\frac{1}{d} x^2\right)\)[/tex]:
[tex]\[
\log _4\left(\frac{1}{d} x^2\right)
\][/tex]
Using the property of logarithms that [tex]\(\log_b(\frac{a}{c}) = \log_b(a) - \log_b(c)\)[/tex], we get:
[tex]\[
\log _4\left(\frac{1}{d} x^2\right) = \log_4(1 x^2) - \log_4(d) = \log_4(x^2) - \log_4(d)
\][/tex]
Then, using the property that [tex]\(\log_b(a^c) = c \log_b(a)\)[/tex]:
[tex]\[
\log_4(x^2) = 2 \log_4(x)
\][/tex]
So the expression simplifies to:
[tex]\[
\log_4\left(\frac{1}{d} x^2\right) = 2 \log_4(x) - \log_4(d)
\][/tex]
Now let's evaluate each given option to see if it matches [tex]\(2 \log_4(x) - \log_4(d)\)[/tex]:
1. [tex]\(2 \log _4\left(\frac{1}{4}\right)-\log _4 x^2\)[/tex]
[tex]\[
2 \log_4\left(\frac{1}{4}\right) = 2 \cdot (-1) = -2, \quad \log_4(x^2) = 2 \log_4(x)
\][/tex]
[tex]\[
2 \log_4\left(\frac{1}{4}\right) - \log_4(x^2) = -2 - 2 \log_4(x)
\][/tex]
This does not match [tex]\(2 \log_4(x) - \log_4(d)\)[/tex].
2. [tex]\(-1 + 2 \log_4(x)\)[/tex]
This form does not account for [tex]\(- \log_4(d)\)[/tex] in [tex]\(2 \log_4(x) - \log_4(d)\)[/tex].
3. [tex]\(-2 + 2 \log_4(x)\)[/tex]
Similar to option 2, this form does not account for [tex]\(- \log_4(d)\)[/tex].
4. [tex]\(2 \log_4\left(\frac{1}{4} x\right)\)[/tex]
[tex]\[
2 \log_4\left(\frac{1}{4} x\right) = 2 \left(\log_4\left(\frac{1}{4}\right) + \log_4(x)\right)
\][/tex]
[tex]\[
= 2 \left(-1 + \log_4(x)\right) = 2 \cdot -1 + 2 \log_4(x) = -2 + 2 \log_4(x)
\][/tex]
This does not match [tex]\(2 \log_4(x) - \log_4(d)\)[/tex].
5. [tex]\(\log_4\left(\frac{1}{4}\right) + \log_4(x^2)\)[/tex]
[tex]\[
\log_4\left(\frac{1}{4}\right) = -1, \quad \log_4(x^2) = 2 \log_4(x)
\][/tex]
[tex]\[
\log_4\left(\frac{1}{4}\right) + \log_4(x^2) = -1 + 2 \log_4(x)
\][/tex]
This does not match [tex]\(2 \log_4(x) - \log_4(d)\)[/tex].
Based on the detailed analysis, none of the given expressions are equivalent to [tex]\(\log _4\left(\frac{1}{d} x^2\right)\)[/tex].
The correct answer is: [tex]\([]\)[/tex] (none of the options provided are correct).
