log3^(3x-6)>2 inequation

Given the inequality:
$$\log{\left(3 \right)}^{3 x - 6} > 2$$
To solve this inequality, we must first solve the corresponding equation:
$$\log{\left(3 \right)}^{3 x - 6} = 2$$
Solve:
Given the equation:
$$\log{\left(3 \right)}^{3 x - 6} = 2$$
or
$$\log{\left(3 \right)}^{3 x - 6} - 2 = 0$$
or
$$\frac{\log{\left(3 \right)}^{3 x}}{\log{\left(3 \right)}^{6}} = 2$$
or
$$\log{\left(3 \right)}^{3 x} = 2 \log{\left(3 \right)}^{6}$$
- this is the simplest exponential equation
Do replacement
$$v = \log{\left(3 \right)}^{3 x}$$
we get
$$v - 2 \log{\left(3 \right)}^{6} = 0$$
or
$$v - 2 \log{\left(3 \right)}^{6} = 0$$
Expand brackets in the left part

v - 2*log3^6 = 0

Divide both parts of the equation by (v - 2*log(3)^6)/v

v = 0 / ((v - 2*log(3)^6)/v)

do backward replacement
$$\log{\left(3 \right)}^{3 x} = v$$
or
$$x = \frac{\log{\left(v \right)}}{\log{\left(\log{\left(3 \right)}^{3} \right)}}$$
$$x_{1} = 2 \log{\left(3 \right)}^{6}$$
$$x_{1} = 2 \log{\left(3 \right)}^{6}$$
This roots
$$x_{1} = 2 \log{\left(3 \right)}^{6}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + 2 \log{\left(3 \right)}^{6}$$
=
$$- \frac{1}{10} + 2 \log{\left(3 \right)}^{6}$$
substitute to the expression
$$\log{\left(3 \right)}^{3 x - 6} > 2$$
$$\log{\left(3 \right)}^{-6 + 3 \left(- \frac{1}{10} + 2 \log{\left(3 \right)}^{6}\right)} > 2$$

          63        6       
        - -- + 6*log (3)    
          10             > 2
(log(3))                    
    

Then
$$x < 2 \log{\left(3 \right)}^{6}$$
no execute
the solution of our inequality is:
$$x > 2 \log{\left(3 \right)}^{6}$$

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