log3x<5 inequation

Given the inequality:
$$\log{\left(3 x \right)} < 5$$
To solve this inequality, we must first solve the corresponding equation:
$$\log{\left(3 x \right)} = 5$$
Solve:
Given the equation
$$\log{\left(3 x \right)} = 5$$
$$\log{\left(3 x \right)} = 5$$
This equation is of the form:

log(v)=p

By definition log

v=e^p

then
$$3 x = e^{\frac{5}{1}}$$
simplify
$$3 x = e^{5}$$
$$x = \frac{e^{5}}{3}$$
$$x_{1} = \frac{e^{5}}{3}$$
$$x_{1} = \frac{e^{5}}{3}$$
This roots
$$x_{1} = \frac{e^{5}}{3}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + \frac{e^{5}}{3}$$
=
$$- \frac{1}{10} + \frac{e^{5}}{3}$$
substitute to the expression
$$\log{\left(3 x \right)} < 5$$
$$\log{\left(3 \left(- \frac{1}{10} + \frac{e^{5}}{3}\right) \right)} < 5$$

   /  3     5\    
log|- -- + e | < 5
   \  10     /    

the solution of our inequality is:
$$x < \frac{e^{5}}{3}$$

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