Solve for all values of x
log, base, 2, left parenthesis, 2, x, squared, minus, 2, x, right parenthesis, equals, 2
log 
2
​	
 (2x 
2
 −2x)=
2

We are given the equation:

\[
\log_2(2x^2 - 2x) = 2
\]

### Step 1: Convert the logarithmic equation to an exponential form.

Using the logarithmic identity:

\[
\log_b(A) = C \quad \text{is equivalent to} \quad A = b^C
\]

In this case, the base \(b = 2\), and the equation becomes:

\[
2x^2 - 2x = 2^2
\]

\[
2x^2 - 2x = 4
\]

### Step 2: Set the equation equal to zero.

Rearrange the equation to form a standard quadratic:

\[
2x^2 - 2x - 4 = 0
\]

### Step 3: Simplify the quadratic equation.

Divide the entire equation by 2 to simplify:

\[
x^2 - x - 2 = 0
\]

### Step 4: Factor the quadratic.

Factor the equation:

\[
(x - 2)(x + 1) = 0
\]

### Step 5: Solve for \(x\).

Set each factor equal to zero:

\[
x - 2 = 0 \quad \Rightarrow \quad x = 2
\]

\[
x + 1 = 0 \quad \Rightarrow \quad x = -1
\]

### Step 6: Check for extraneous solutions.

Since we are solving a logarithmic equation, we must ensure that the arguments of the logarithm are positive (the domain of the logarithm). The argument of the logarithm is \(2x^2 - 2x\). Substituting \(x = 2\) and \(x = -1\):

- For \(x = 2\):

\[
2(2)^2 - 2(2) = 8 - 4 = 4 \quad (\text{positive})
\]

- For \(x = -1\):

\[
2(-1)^2 - 2(-1) = 2 + 2 = 4 \quad (\text{positive})
\]

Both values of \(x\) satisfy the domain of the logarithmic function.

### Final Answer:
The solutions are:

\[
x = 2 \quad \text{and} \quad x = -1
\]

Would you like more details or have any questions about the steps?

Here are 5 related questions for further practice:
1. Solve \(\log_2(x^2 - 3x) = 1\).
2. Solve \(\log_3(3x^2 - 6x) = 2\).
3. Find the domain of \(\log_2(3x^2 - x)\).
4. Solve \(\log_4(x^2 - 2x + 1) = 0\).
5. Solve \(\log_5(x^2 - 4x + 4) = 1\).

**Tip:** Always check if the arguments inside logarithmic functions are positive when solving logarithmic equations to avoid extraneous solutions.

Solve for all values of x: log_2(2x^2 - 2x) = 2

Learn how to solve the logarithmic equation log_2(2x^2 - 2x) = 2 with a step-by-step solution. This problem involves converting logarithmic to exponential form and solving quadratic equations. Suitable for students in Grades 9-12.

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