Mister Exam
log33x<2 inequation
The solution
Detail solution
Given the inequality:
$$\log{\left(33 x \right)} < 2$$
To solve this inequality, we must first solve the corresponding equation:
$$\log{\left(33 x \right)} = 2$$
Solve:
Given the equation
$$\log{\left(33 x \right)} = 2$$
$$\log{\left(33 x \right)} = 2$$
This equation is of the form:
By definition log
then
$$33 x = e^{\frac{2}{1}}$$
simplify
$$33 x = e^{2}$$
$$x = \frac{e^{2}}{33}$$
$$x_{1} = \frac{e^{2}}{33}$$
$$x_{1} = \frac{e^{2}}{33}$$
This roots
$$x_{1} = \frac{e^{2}}{33}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + \frac{e^{2}}{33}$$
=
$$- \frac{1}{10} + \frac{e^{2}}{33}$$
substitute to the expression
$$\log{\left(33 x \right)} < 2$$
$$\log{\left(33 \left(- \frac{1}{10} + \frac{e^{2}}{33}\right) \right)} < 2$$
the solution of our inequality is:
$$x < \frac{e^{2}}{33}$$
$$\log{\left(33 x \right)} < 2$$
To solve this inequality, we must first solve the corresponding equation:
$$\log{\left(33 x \right)} = 2$$
Solve:
Given the equation
$$\log{\left(33 x \right)} = 2$$
$$\log{\left(33 x \right)} = 2$$
This equation is of the form:
log(v)=p
By definition log
v=e^p
then
$$33 x = e^{\frac{2}{1}}$$
simplify
$$33 x = e^{2}$$
$$x = \frac{e^{2}}{33}$$
$$x_{1} = \frac{e^{2}}{33}$$
$$x_{1} = \frac{e^{2}}{33}$$
This roots
$$x_{1} = \frac{e^{2}}{33}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + \frac{e^{2}}{33}$$
=
$$- \frac{1}{10} + \frac{e^{2}}{33}$$
substitute to the expression
$$\log{\left(33 x \right)} < 2$$
$$\log{\left(33 \left(- \frac{1}{10} + \frac{e^{2}}{33}\right) \right)} < 2$$
/ 33 2\ log|- -- + e | < 2 \ 10 /
the solution of our inequality is:
$$x < \frac{e^{2}}{33}$$
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x1
Solving inequality on a graph
Rapid solution 2
[src]
2
e
(0, --)
33
$$x\ in\ \left(0, \frac{e^{2}}{33}\right)$$
x in Interval.open(0, exp(2)/33)
Rapid solution
[src]
/ 2\ | e | And|0 < x, x < --| \ 33/
$$0 < x \wedge x < \frac{e^{2}}{33}$$
(0 < x)∧(x < exp(2)/33)