We are given the equation:

$2 \cdot \log_2(x) - \log_2(x) - \log_2(2) = 2$

Step 1: Simplify the equation.

First, recall that $\log_2(2) = 1$, since $2^1 = 2$. This simplifies the equation to:

$2 \cdot \log_2(x) - \log_2(x) - 1 = 2$

Step 2: Combine like terms.

Next, combine the logarithmic terms:

$(2\log_2(x) - \log_2(x)) - 1 = 2$

This simplifies to:

$\log_2(x) - 1 = 2$

Step 3: Solve for $\log_2(x)$.

Now add 1 to both sides:

$\log_2(x) = 3$

Step 4: Solve for $x$.

By the definition of a logarithm, $x = 2^3 = 8$.

Final Answer:

$x = 8$

Would you like more details on any step? Here are 5 questions to expand on this:

1. How do logarithmic properties help simplify such equations?
2. Why does $\log_2(2) = 1$?
3. Can we apply the same steps with different logarithmic bases?
4. What other methods could be used to solve logarithmic equations?
5. How does the logarithm change if there's a coefficient in front of it?

Tip: When solving logarithmic equations, always simplify constants like $\log_b(b) = 1$ to make the equation easier to manage.