logx(3)<0 inequation

Given the inequality:
$$3 \log{\left(x \right)} < 0$$
To solve this inequality, we must first solve the corresponding equation:
$$3 \log{\left(x \right)} = 0$$
Solve:
Given the equation
$$3 \log{\left(x \right)} = 0$$
$$3 \log{\left(x \right)} = 0$$
Let's divide both parts of the equation by the multiplier of log =3
$$\log{\left(x \right)} = 0$$
This equation is of the form:

log(v)=p

By definition log

v=e^p

then
$$x = e^{\frac{0}{3}}$$
simplify
$$x = 1$$
$$x_{1} = 1$$
$$x_{1} = 1$$
This roots
$$x_{1} = 1$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + 1$$
=
$$\frac{9}{10}$$
substitute to the expression
$$3 \log{\left(x \right)} < 0$$
$$3 \log{\left(\frac{9}{10} \right)} < 0$$

3*log(9/10) < 0

the solution of our inequality is:
$$x < 1$$

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