Let's solve the given problem:

The problem states:

$x = 4^{5 \cdot \log_4 6 - \log_4 7}.$

Step 1: Use the properties of logarithms

We know the logarithmic property: $a^{\log_a b} = b.$

Additionally, the subtraction rule for logarithms states: $\log_a b - \log_a c = \log_a \left(\frac{b}{c}\right).$

So, rewrite the exponent: $5 \cdot \log_4 6 - \log_4 7 = \log_4 (6^5) - \log_4 7 = \log_4 \left(\frac{6^5}{7}\right).$

Step 2: Simplify the expression for $x$

Now, substitute the simplified exponent into the original equation: $x = 4^{\log_4 \left(\frac{6^5}{7}\right)}.$

Using $a^{\log_a b} = b$, this simplifies to: $x = \frac{6^5}{7}.$

Step 3: Compute $6^5$

$6^5 = 6 \cdot 6 \cdot 6 \cdot 6 \cdot 6 = 7776.$

Thus: $x = \frac{7776}{7}.$

Final Answer:

$x = \frac{7776}{7}.$

Let me know if you need further clarification or additional details!

Here are 5 related questions for you to practice:

1. Solve $x = 3^{2 \log_3 5 - \log_3 2}$ and simplify.
2. Simplify $x = 10^{\log_{10} 25 + \log_{10} 4}$.
3. Calculate $x = 2^{4 \log_2 3 + \log_2 5}$.
4. Use logarithmic properties to expand $\log_5 (125 \cdot 4 / 2)$.
5. Simplify $x = 8^{\log_8 (16)}$.

Tip: Always look for ways to simplify exponents and logarithmic expressions using their properties!