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.

Explore how to solve the logarithmic equation 2xlog₂ - log₂x - log₂ = 2 through simplification and properties of logarithms. This detailed solution is suitable for students in Grades 9-11, covering key concepts like logarithmic properties and algebraic manipulation.

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