log0,5x>0 inequation

Given the inequality:
$$\log{\left(0.5 x \right)} > 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\log{\left(0.5 x \right)} = 0$$
Solve:
Given the equation
$$\log{\left(0.5 x \right)} = 0$$
$$\log{\left(0.5 x \right)} = 0$$
This equation is of the form:

log(v)=p

By definition log

v=e^p

then
$$0.5 x = e^{\frac{0}{1}}$$
simplify
$$0.5 x = 1$$
$$x = 2$$
$$x_{1} = 2$$
$$x_{1} = 2$$
This roots
$$x_{1} = 2$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + 2$$
=
$$1.9$$
substitute to the expression
$$\log{\left(0.5 x \right)} > 0$$
$$\log{\left(0.5 \cdot 1.9 \right)} > 0$$

-0.0512932943875506 > 0

Then
$$x < 2$$
no execute
the solution of our inequality is:
$$x > 2$$

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