The given expression is:

$2 \log_6 4 + 3 \log_6 x - 4 \log_6 y$

We can simplify this by using the logarithmic properties:

1. Power Rule: $\log_b (a^n) = n \log_b a$.
2. Product Rule: $\log_b a + \log_b b = \log_b (a \cdot b)$.
3. Quotient Rule: $\log_b a - \log_b b = \log_b \left(\frac{a}{b}\right)$.

Step-by-step simplification:

1. Apply the Power Rule: $2 \log_6 4 = \log_6 4^2 = \log_6 16$ $3 \log_6 x = \log_6 x^3$ $4 \log_6 y = \log_6 y^4$

2. Substitute these into the original expression: $\log_6 16 + \log_6 x^3 - \log_6 y^4$

3. Apply the Product Rule to the first two terms: $\log_6 (16 \cdot x^3) - \log_6 y^4$

4. Finally, apply the Quotient Rule: $\log_6 \left( \frac{16 \cdot x^3}{y^4} \right)$

Final simplified expression:

$\log_6 \left( \frac{16x^3}{y^4} \right)$

Would you like any additional details or further explanation?

Here are 5 related questions for practice:

1. Simplify $3 \log_2 5 + \log_2 8 - 4 \log_2 2$.
2. Solve $\log_3 (x^2 + 3x - 4) = 2$.
3. Expand $\log_b (a \cdot b^2)$ using logarithmic properties.
4. Simplify $\log_7 49 + \log_7 x - \log_7 y^2$.
5. Solve for $x$ in $2 \log_4 x = 3$.

Tip: Always keep logarithmic properties in mind to simplify complex expressions easily!