Solve the logarithmic equation: 2 x − 2(x − 2) = 3

The definition of a logarithm states that $\log_b a = c$ is equivalent to $b^c = a$.
Applying this to our equation, where $b=2$, $a=\frac{x}{x-2}$, and $c=3$:

$$2^3 = \frac{x}{x - 2}$$

$$8 = \frac{x}{x - 2}$$

Multiply both sides by $(x - 2)$:

$$8(x - 2) = x$$ $$8x - 16 = x$$

Subtract $x$ from both sides:

$$7x - 16 = 0$$

Add 16 to both sides:

$$7x = 16$$

Divide by 7:

$$x = \frac{16}{7}$$

The arguments of the original logarithms must be positive.
For $\log_2 x$, we need $x > 0$. Our solution $x = \frac{16}{7}$ satisfies this condition.
For $\log_2(x - 2)$, we need $x - 2 > 0$, which means $x > 2$.
Our solution $x = \frac{16}{7} \approx 2.286$, which is greater than 2.
Both conditions are satisfied, so the solution is valid.

The value of $x$ is $\boxed{\frac{16}{7}}$.