Mister Exam
log^23x-2log3x<=3 inequation
The solution
You have entered
[src]
23 log (x) - 2*log(3*x) <= 3
$$\log{\left(x \right)}^{23} - 2 \log{\left(3 x \right)} \leq 3$$
log(x)^23 - 2*log(3*x) <= 3
Detail solution
Given the inequality:
$$\log{\left(x \right)}^{23} - 2 \log{\left(3 x \right)} \leq 3$$
To solve this inequality, we must first solve the corresponding equation:
$$\log{\left(x \right)}^{23} - 2 \log{\left(3 x \right)} = 3$$
Solve:
$$x_{1} = 2.97660479160336$$
$$x_{1} = 2.97660479160336$$
This roots
$$x_{1} = 2.97660479160336$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} \leq x_{1}$$
For example, let's take the point
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + 2.97660479160336$$
=
$$2.87660479160336$$
substitute to the expression
$$\log{\left(x \right)}^{23} - 2 \log{\left(3 x \right)} \leq 3$$
$$- 2 \log{\left(2.87660479160336 \cdot 3 \right)} + \log{\left(2.87660479160336 \right)}^{23} \leq 3$$
the solution of our inequality is:
$$x \leq 2.97660479160336$$
$$\log{\left(x \right)}^{23} - 2 \log{\left(3 x \right)} \leq 3$$
To solve this inequality, we must first solve the corresponding equation:
$$\log{\left(x \right)}^{23} - 2 \log{\left(3 x \right)} = 3$$
Solve:
$$x_{1} = 2.97660479160336$$
$$x_{1} = 2.97660479160336$$
This roots
$$x_{1} = 2.97660479160336$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} \leq x_{1}$$
For example, let's take the point
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + 2.97660479160336$$
=
$$2.87660479160336$$
substitute to the expression
$$\log{\left(x \right)}^{23} - 2 \log{\left(3 x \right)} \leq 3$$
$$- 2 \log{\left(2.87660479160336 \cdot 3 \right)} + \log{\left(2.87660479160336 \right)}^{23} \leq 3$$
-0.761943084108506 <= 3
the solution of our inequality is:
$$x \leq 2.97660479160336$$
_____
\
-------•-------
x1
Solving inequality on a graph