We are tasked with evaluating:

$\log_4 \left( \frac{1}{256} \right)$

Step 1: Express 256 as a power of 4

First, recognize that $256$ can be written as a power of 4. To do this, observe that:

$256 = 4^4$

Step 2: Simplify the logarithm expression

Now we can substitute $256$ with $4^4$:

$\log_4 \left( \frac{1}{4^4} \right)$

Using the property of exponents that $\frac{1}{a^n} = a^{-n}$, we rewrite the expression:

$\log_4 \left( 4^{-4} \right)$

Step 3: Apply the logarithmic identity

We know that $\log_b (b^x) = x$. So:

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

Thus, the value of the logarithm is:

$\boxed{-4}$

Let me know if you'd like further details or have any questions.

Here are 5 related questions to expand your understanding:

1. How do you simplify $\log_4 (64)$?
2. What is $\log_4 (1/64)$?
3. How do you solve $\log_b (b^x) = x$ for a general base $b$?
4. Can you express $\log_4 (1/1024)$ in terms of powers of 4?
5. How does the change of base formula work for logarithms?

Tip: Remember that logarithms and exponents are inverse operations. If you see an exponent inside a logarithm, you can often use this relationship to simplify.