Solve: (ln x)² - (ln 2)(ln x) < 2(ln 2)²

To solve the inequality \((\ln x)^2 - (\ln 2)(\ln x) < 2(\ln 2)^2\), follow these steps:

### Step 1: Let \( y = \ln x \)
Substitute \( y = \ln x \) to simplify the equation. The inequality becomes:
\[
y^2 - (\ln 2)y < 2(\ln 2)^2
\]

### Step 2: Rearrange into standard quadratic form
Rewriting the inequality:
\[
y^2 - (\ln 2)y - 2(\ln 2)^2 < 0
\]

### Step 3: Solve the quadratic equation
To solve the quadratic equation \( y^2 - (\ln 2)y - 2(\ln 2)^2 = 0 \), use the quadratic formula:
\[
y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
Here:
- \( a = 1 \),
- \( b = -\ln 2 \),
- \( c = -2(\ln 2)^2 \).

Substitute these values:
\[
y = \frac{-(-\ln 2) \pm \sqrt{(-\ln 2)^2 - 4(1)(-2(\ln 2)^2)}}{2(1)}
\]
\[
y = \frac{\ln 2 \pm \sqrt{(\ln 2)^2 + 8(\ln 2)^2}}{2}
\]
\[
y = \frac{\ln 2 \pm \sqrt{9(\ln 2)^2}}{2}
\]
\[
y = \frac{\ln 2 \pm 3\ln 2}{2}
\]

### Step 4: Find the roots
Split the solution into two cases:
1. \( y = \frac{\ln 2 + 3\ln 2}{2} = \frac{4\ln 2}{2} = 2\ln 2 \)
2. \( y = \frac{\ln 2 - 3\ln 2}{2} = \frac{-2\ln 2}{2} = -\ln 2 \)

The roots are \( y = 2\ln 2 \) and \( y = -\ln 2 \).

### Step 5: Analyze the inequality
The quadratic inequality \( y^2 - (\ln 2)y - 2(\ln 2)^2 < 0 \) represents a parabola opening upwards (since the coefficient of \( y^2 \) is positive). The solution to the inequality lies between the roots:
\[
-\ln 2 < y < 2\ln 2
\]

### Step 6: Substitute back \( y = \ln x \)
Returning to the original variable \( x \):
\[
-\ln 2 < \ln x < 2\ln 2
\]

Exponentiate both sides to solve for \( x \):
\[
e^{-\ln 2} < x < e^{2\ln 2}
\]
\[
\frac{1}{2} < x < 4
\]

### Final Answer
The solution to the inequality is:
\[
\frac{1}{2} < x < 4
\]