logx(x-2)×logx(x+2)<0 inequation

Given the inequality:
$$\left(x + 2\right) \left(x - 2\right) \log{\left(x \right)} \log{\left(x \right)} < 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\left(x + 2\right) \left(x - 2\right) \log{\left(x \right)} \log{\left(x \right)} = 0$$
Solve:
Given the equation
$$\left(x + 2\right) \left(x - 2\right) \log{\left(x \right)} \log{\left(x \right)} = 0$$
transform
$$\left(x^{2} - 4\right) \log{\left(x \right)}^{2} = 0$$
$$\left(x + 2\right) \left(x - 2\right) \log{\left(x \right)} \log{\left(x \right)} + 0 = 0$$
Do replacement
$$w = \log{\left(x \right)}$$
Expand the expression in the equation
$$w^{2} \left(x - 2\right) \left(x + 2\right) = 0$$
We get the quadratic equation
$$w^{2} x^{2} - 4 w^{2} = 0$$
This equation is of the form

a*w^2 + b*w + c = 0

A quadratic equation can be solved
using the discriminant.
The roots of the quadratic equation:
$$w_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$w_{2} = \frac{- \sqrt{D} - b}{2 a}$$
where D = b^2 - 4*a*c - it is the discriminant.
Because
$$a = x^{2} - 4$$
$$b = 0$$
$$c = 0$$
, then

D = b^2 - 4 * a * c = 

(0)^2 - 4 * (-4 + x^2) * (0) = 0

Because D = 0, then the equation has one root.

w = -b/2a = -0/2/(-4 + x^2)

$$w_{1} = 0$$
do backward replacement
$$\log{\left(x \right)} = w$$
Given the equation
$$\log{\left(x \right)} = w$$
$$\log{\left(x \right)} = w$$
This equation is of the form:

log(v)=p

By definition log

v=e^p

then
$$1 x + 0 = e^{\frac{w}{1}}$$
simplify
$$x = e^{w}$$
substitute w:
$$x_{1} = -2$$
$$x_{2} = 1$$
$$x_{3} = 2$$
$$x_{1} = -2$$
$$x_{2} = 1$$
$$x_{3} = 2$$
This roots
$$x_{1} = -2$$
$$x_{2} = 1$$
$$x_{3} = 2$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} < x_{1}$$
For example, let's take the point
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$-2 - \frac{1}{10}$$
=
$$- \frac{21}{10}$$
substitute to the expression
$$\left(x + 2\right) \left(x - 2\right) \log{\left(x \right)} \log{\left(x \right)} < 0$$
$$\left(- \frac{21}{10} + 2\right) \left(- \frac{21}{10} - 2\right) \log{\left(- \frac{21}{10} \right)} \log{\left(- \frac{21}{10} \right)} < 0$$

                   2    
   /          /21\\     
41*|pi*I + log|--||     
   \          \10//  < 0
--------------------    
        100             
    

Then
$$x < -2$$
no execute
one of the solutions of our inequality is:
$$x > -2 \wedge x < 1$$

         _____           _____  
        /     \         /
-------ο-------ο-------ο-------
       x_1      x_2      x_3

Other solutions will get with the changeover to the next point
etc.
The answer:
$$x > -2 \wedge x < 1$$
$$x > 2$$