To simplify the logarithmic expression
$\log_5 \left( \frac{\sqrt[6]{4x + 1}}{4x^2} \right),$ we will apply the logarithmic rules step-by-step:

1. Quotient Rule: $\log_b \left( \frac{A}{B} \right) = \log_b (A) - \log_b (B)$.
2. Power Rule: $\log_b (A^n) = n \log_b (A)$.

Step-by-step Solution

1. Start with the given expression:
   $\log_5 \left( \frac{\sqrt[6]{4x + 1}}{4x^2} \right).$
   Apply the quotient rule:
   [ \log_5 \left( \frac{\sqrt[6]{4x + 1}}{4x^2} \right) = \log_5 (\sqrt[6]{4x + 1}) - \log_5 (4x^2). ]

2. Simplify each term:

   • For $\log_5 (\sqrt[6]{4x + 1})$, use the power rule:
     $\log_5 (\sqrt[6]{4x + 1}) = \frac{1}{6} \log_5 (4x + 1).$

   • For $\log_5 (4x^2)$, rewrite as a product:
     $\log_5 (4x^2) = \log_5 (4) + \log_5 (x^2).$
     Apply the power rule to $\log_5 (x^2)$:
     $\log_5 (x^2) = 2 \log_5 (x).$
     Thus:
     $\log_5 (4x^2) = \log_5 (4) + 2 \log_5 (x).$

3. Substitute back into the original expression:
   [ \log_5 \left( \frac{\sqrt[6]{4x + 1}}{4x^2} \right) = \frac{1}{6} \log_5 (4x + 1) - \left[ \log_5 (4) + 2 \log_5 (x) \right]. ]

4. Simplify further:
   Distribute the negative sign:
   $\log_5 \left( \frac{\sqrt[6]{4x + 1}}{4x^2} \right) = \frac{1}{6} \log_5 (4x + 1) - \log_5 (4) - 2 \log_5 (x).$

Final Answer:

The correct option is:
$\log_5 \left( \frac{\sqrt[6]{4x + 1}}{4x^2} \right) = \frac{1}{6} \log_5 (4x + 1) - 2 \log_5 (x) - \log_5 (4).$

This matches Option 1.

Do you need further clarification?

Here are 5 related questions you might find useful:

1. How do you apply the quotient rule in logarithms?
2. What is the power rule for logarithms and how is it used?
3. How can you rewrite logarithms of products and powers?
4. What is the significance of simplifying logarithmic expressions?
5. Can you explain the steps for simplifying a nested logarithmic expression?

Tip: Always rewrite roots as fractional powers before applying logarithmic rules. This simplifies calculations greatly!