Mister Exam
log⅓2*log5(x-2)>0 inequation
The solution
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log(x - 2)
log(1/3)*2*---------- > 0
log(5)
$$2 \log{\left(\frac{1}{3} \right)} \frac{\log{\left(x - 2 \right)}}{\log{\left(5 \right)}} > 0$$
(2*log(1/3))*(log(x - 2)/log(5)) > 0
Detail solution
Given the inequality:
$$2 \log{\left(\frac{1}{3} \right)} \frac{\log{\left(x - 2 \right)}}{\log{\left(5 \right)}} > 0$$
To solve this inequality, we must first solve the corresponding equation:
$$2 \log{\left(\frac{1}{3} \right)} \frac{\log{\left(x - 2 \right)}}{\log{\left(5 \right)}} = 0$$
Solve:
Given the equation
$$2 \log{\left(\frac{1}{3} \right)} \frac{\log{\left(x - 2 \right)}}{\log{\left(5 \right)}} = 0$$
$$- \frac{2 \log{\left(3 \right)} \log{\left(x - 2 \right)}}{\log{\left(5 \right)}} = 0$$
Let's divide both parts of the equation by the multiplier of log =-2*log(3)/log(5)
$$\log{\left(x - 2 \right)} = 0$$
This equation is of the form:
By definition log
then
$$x - 2 = e^{\frac{0}{\left(-1\right) 2 \log{\left(3 \right)} \frac{1}{\log{\left(5 \right)}}}}$$
simplify
$$x - 2 = 1$$
$$x = 3$$
$$x_{1} = 3$$
$$x_{1} = 3$$
This roots
$$x_{1} = 3$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + 3$$
=
$$\frac{29}{10}$$
substitute to the expression
$$2 \log{\left(\frac{1}{3} \right)} \frac{\log{\left(x - 2 \right)}}{\log{\left(5 \right)}} > 0$$
$$2 \log{\left(\frac{1}{3} \right)} \frac{\log{\left(-2 + \frac{29}{10} \right)}}{\log{\left(5 \right)}} > 0$$
the solution of our inequality is:
$$x < 3$$
$$2 \log{\left(\frac{1}{3} \right)} \frac{\log{\left(x - 2 \right)}}{\log{\left(5 \right)}} > 0$$
To solve this inequality, we must first solve the corresponding equation:
$$2 \log{\left(\frac{1}{3} \right)} \frac{\log{\left(x - 2 \right)}}{\log{\left(5 \right)}} = 0$$
Solve:
Given the equation
$$2 \log{\left(\frac{1}{3} \right)} \frac{\log{\left(x - 2 \right)}}{\log{\left(5 \right)}} = 0$$
$$- \frac{2 \log{\left(3 \right)} \log{\left(x - 2 \right)}}{\log{\left(5 \right)}} = 0$$
Let's divide both parts of the equation by the multiplier of log =-2*log(3)/log(5)
$$\log{\left(x - 2 \right)} = 0$$
This equation is of the form:
log(v)=p
By definition log
v=e^p
then
$$x - 2 = e^{\frac{0}{\left(-1\right) 2 \log{\left(3 \right)} \frac{1}{\log{\left(5 \right)}}}}$$
simplify
$$x - 2 = 1$$
$$x = 3$$
$$x_{1} = 3$$
$$x_{1} = 3$$
This roots
$$x_{1} = 3$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + 3$$
=
$$\frac{29}{10}$$
substitute to the expression
$$2 \log{\left(\frac{1}{3} \right)} \frac{\log{\left(x - 2 \right)}}{\log{\left(5 \right)}} > 0$$
$$2 \log{\left(\frac{1}{3} \right)} \frac{\log{\left(-2 + \frac{29}{10} \right)}}{\log{\left(5 \right)}} > 0$$
-2*log(3)*log(9/10)
------------------- > 0
log(5) the solution of our inequality is:
$$x < 3$$
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