Mister Exam
log3(5x-9)<=20 inequation
The solution
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log(5*x - 9) ------------ <= 20 log(3)
$$\frac{\log{\left(5 x - 9 \right)}}{\log{\left(3 \right)}} \leq 20$$
log(5*x - 9)/log(3) <= 20
Detail solution
Given the inequality:
$$\frac{\log{\left(5 x - 9 \right)}}{\log{\left(3 \right)}} \leq 20$$
To solve this inequality, we must first solve the corresponding equation:
$$\frac{\log{\left(5 x - 9 \right)}}{\log{\left(3 \right)}} = 20$$
Solve:
Given the equation
$$\frac{\log{\left(5 x - 9 \right)}}{\log{\left(3 \right)}} = 20$$
$$\frac{\log{\left(5 x - 9 \right)}}{\log{\left(3 \right)}} = 20$$
Let's divide both parts of the equation by the multiplier of log =1/log(3)
$$\log{\left(5 x - 9 \right)} = 20 \log{\left(3 \right)}$$
This equation is of the form:
By definition log
then
$$5 x - 9 = e^{\frac{20}{\frac{1}{\log{\left(3 \right)}}}}$$
simplify
$$5 x - 9 = 3486784401$$
$$5 x = 3486784410$$
$$x = 697356882$$
$$x_{1} = 697356882$$
$$x_{1} = 697356882$$
This roots
$$x_{1} = 697356882$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + 697356882$$
=
$$\frac{6973568819}{10}$$
substitute to the expression
$$\frac{\log{\left(5 x - 9 \right)}}{\log{\left(3 \right)}} \leq 20$$
$$\frac{\log{\left(-9 + \frac{5 \cdot 6973568819}{10} \right)}}{\log{\left(3 \right)}} \leq 20$$
the solution of our inequality is:
$$x \leq 697356882$$
$$\frac{\log{\left(5 x - 9 \right)}}{\log{\left(3 \right)}} \leq 20$$
To solve this inequality, we must first solve the corresponding equation:
$$\frac{\log{\left(5 x - 9 \right)}}{\log{\left(3 \right)}} = 20$$
Solve:
Given the equation
$$\frac{\log{\left(5 x - 9 \right)}}{\log{\left(3 \right)}} = 20$$
$$\frac{\log{\left(5 x - 9 \right)}}{\log{\left(3 \right)}} = 20$$
Let's divide both parts of the equation by the multiplier of log =1/log(3)
$$\log{\left(5 x - 9 \right)} = 20 \log{\left(3 \right)}$$
This equation is of the form:
log(v)=p
By definition log
v=e^p
then
$$5 x - 9 = e^{\frac{20}{\frac{1}{\log{\left(3 \right)}}}}$$
simplify
$$5 x - 9 = 3486784401$$
$$5 x = 3486784410$$
$$x = 697356882$$
$$x_{1} = 697356882$$
$$x_{1} = 697356882$$
This roots
$$x_{1} = 697356882$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + 697356882$$
=
$$\frac{6973568819}{10}$$
substitute to the expression
$$\frac{\log{\left(5 x - 9 \right)}}{\log{\left(3 \right)}} \leq 20$$
$$\frac{\log{\left(-9 + \frac{5 \cdot 6973568819}{10} \right)}}{\log{\left(3 \right)}} \leq 20$$
log(6973568801/2)
----------------- <= 20
log(3) the solution of our inequality is:
$$x \leq 697356882$$
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