The expression given in the image is:

$\log_4 \left( \frac{4}{x} \right)^2$

We will expand this logarithmic expression step by step.

Step 1: Apply the Power Rule of Logarithms

The power rule states that:

$\log_b (M^n) = n \cdot \log_b M$

In our case, $M = \frac{4}{x}$ and $n = 2$, so:

$\log_4 \left( \frac{4}{x} \right)^2 = 2 \cdot \log_4 \left( \frac{4}{x} \right)$

Step 2: Apply the Quotient Rule of Logarithms

The quotient rule states that:

$\log_b \left( \frac{M}{N} \right) = \log_b M - \log_b N$

Here, $M = 4$ and $N = x$, so:

$\log_4 \left( \frac{4}{x} \right) = \log_4 4 - \log_4 x$

Since $\log_4 4 = 1$ (because $4^1 = 4$), we get:

$\log_4 \left( \frac{4}{x} \right) = 1 - \log_4 x$

Step 3: Substitute Back

Now substitute this result back into the expression from Step 1:

$2 \cdot \log_4 \left( \frac{4}{x} \right) = 2 \cdot (1 - \log_4 x)$

Step 4: Simplify

Distribute the 2:

$2 \cdot (1 - \log_4 x) = 2 - 2 \log_4 x$

Thus, the expanded form of the expression is:

$\boxed{2 - 2 \log_4 x}$

Let me know if you'd like further clarification or details!

Related Questions:

1. What is the power rule for logarithms and how is it applied?
2. How does the quotient rule for logarithms work with fractions?
3. Why is $\log_4 4 = 1$ and how does this generalize for other bases?
4. How can we apply the logarithm expansion rules in reverse to condense expressions?
5. What is the difference between common logarithms (base 10) and natural logarithms (base $e$)?

Tip:

When expanding or simplifying logarithmic expressions, always check for opportunities to apply rules like the product, quotient, or power rule!