log6x>0 inequation

Given the inequality:
$$\log{\left(6 x \right)} > 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\log{\left(6 x \right)} = 0$$
Solve:
Given the equation
$$\log{\left(6 x \right)} = 0$$
$$\log{\left(6 x \right)} = 0$$
This equation is of the form:

log(v)=p

By definition log

v=e^p

then
$$6 x = e^{\frac{0}{1}}$$
simplify
$$6 x = 1$$
$$x = \frac{1}{6}$$
$$x_{1} = \frac{1}{6}$$
$$x_{1} = \frac{1}{6}$$
This roots
$$x_{1} = \frac{1}{6}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + \frac{1}{6}$$
=
$$\frac{1}{15}$$
substitute to the expression
$$\log{\left(6 x \right)} > 0$$
$$\log{\left(\frac{6}{15} \right)} > 0$$

log(2/5) > 0

Then
$$x < \frac{1}{6}$$
no execute
the solution of our inequality is:
$$x > \frac{1}{6}$$

         _____  
        /
-------ο-------
       x1