log⅓(x²-5x+7)>0 inequation

Given the inequality:
$$\left(\left(x^{2} - 5 x\right) + 7\right) \log{\left(\frac{1}{3} \right)} > 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\left(\left(x^{2} - 5 x\right) + 7\right) \log{\left(\frac{1}{3} \right)} = 0$$
Solve:
Expand the expression in the equation
$$\left(\left(x^{2} - 5 x\right) + 7\right) \log{\left(\frac{1}{3} \right)} = 0$$
We get the quadratic equation
$$- x^{2} \log{\left(3 \right)} + 5 x \log{\left(3 \right)} - 7 \log{\left(3 \right)} = 0$$
This equation is of the form

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

A quadratic equation can be solved
using the discriminant.
The roots of the quadratic equation:
$$x_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$x_{2} = \frac{- \sqrt{D} - b}{2 a}$$
where D = b^2 - 4*a*c - it is the discriminant.
Because
$$a = - \log{\left(3 \right)}$$
$$b = 5 \log{\left(3 \right)}$$
$$c = - 7 \log{\left(3 \right)}$$
, then

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

(5*log(3))^2 - 4 * (-log(3)) * (-7*log(3)) = -3*log(3)^2

Because D<0, then the equation
has no real roots,
but complex roots is exists.

x1 = (-b + sqrt(D)) / (2*a)

x2 = (-b - sqrt(D)) / (2*a)

or
$$x_{1} = - \frac{- 5 \log{\left(3 \right)} + \sqrt{3} i \log{\left(3 \right)}}{2 \log{\left(3 \right)}}$$
$$x_{2} = - \frac{- 5 \log{\left(3 \right)} - \sqrt{3} i \log{\left(3 \right)}}{2 \log{\left(3 \right)}}$$
$$x_{1} = - \frac{- 5 \log{\left(3 \right)} + \sqrt{3} i \log{\left(3 \right)}}{2 \log{\left(3 \right)}}$$
$$x_{2} = - \frac{- 5 \log{\left(3 \right)} - \sqrt{3} i \log{\left(3 \right)}}{2 \log{\left(3 \right)}}$$
Exclude the complex solutions:
This equation has no roots,
this inequality is executed for any x value or has no solutions
check it
subtitute random point x, for example

x0 = 0

$$\left(\left(0^{2} - 0 \cdot 5\right) + 7\right) \log{\left(\frac{1}{3} \right)} > 0$$

-7*log(3) > 0

so the inequality has no solutions