(log(2-5*x)/log(2))>1 inequation

Given the inequality:
$$\frac{\log{\left(2 - 5 x \right)}}{\log{\left(2 \right)}} > 1$$
To solve this inequality, we must first solve the corresponding equation:
$$\frac{\log{\left(2 - 5 x \right)}}{\log{\left(2 \right)}} = 1$$
Solve:
Given the equation
$$\frac{\log{\left(2 - 5 x \right)}}{\log{\left(2 \right)}} = 1$$
$$\frac{\log{\left(2 - 5 x \right)}}{\log{\left(2 \right)}} = 1$$
Let's divide both parts of the equation by the multiplier of log =1/log(2)
$$\log{\left(2 - 5 x \right)} = \log{\left(2 \right)}$$
This equation is of the form:

log(v)=p

By definition log

v=e^p

then
$$2 - 5 x = e^{\frac{1}{\frac{1}{\log{\left(2 \right)}}}}$$
simplify
$$2 - 5 x = 2$$
$$- 5 x = 0$$
$$x = 0$$
$$x_{1} = 0$$
$$x_{1} = 0$$
This roots
$$x_{1} = 0$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10}$$
=
$$- \frac{1}{10}$$
substitute to the expression
$$\frac{\log{\left(2 - 5 x \right)}}{\log{\left(2 \right)}} > 1$$
$$\frac{\log{\left(2 - \frac{\left(-1\right) 5}{10} \right)}}{\log{\left(2 \right)}} > 1$$

log(5/2)    
-------- > 1
 log(2)     

the solution of our inequality is:
$$x < 0$$

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