Mister Exam

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log3(5x-9)<=20 inequation

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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:

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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       x1

Solving inequality on a graph