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Solving inequalities/ 1-sqrt(1-4log4(x)^2)/log4(x)<2

1-sqrt(1-4log4(x)^2)/log4(x)<2 inequation

A inequation with variable

The solution

You have entered [src] 
         _________________    
        /               2     
       /        /log(x)\      
      /   1 - 4*|------|      
    \/          \log(4)/      
1 - ---------------------- < 2
           /log(x)\           
           |------|           
           \log(4)/
$$1 - \frac{\sqrt{1 - 4 \left(\frac{\log{\left(x \right)}}{\log{\left(4 \right)}}\right)^{2}}}{\frac{1}{\log{\left(4 \right)}} \log{\left(x \right)}} < 2$$ 
1 - sqrt(1 - 4*log(x)^2/log(4)^2)/(log(x)/log(4)) < 2
Detail solution
Given the inequality:
$$1 - \frac{\sqrt{1 - 4 \left(\frac{\log{\left(x \right)}}{\log{\left(4 \right)}}\right)^{2}}}{\frac{1}{\log{\left(4 \right)}} \log{\left(x \right)}} < 2$$
To solve this inequality, we must first solve the corresponding equation:
$$1 - \frac{\sqrt{1 - 4 \left(\frac{\log{\left(x \right)}}{\log{\left(4 \right)}}\right)^{2}}}{\frac{1}{\log{\left(4 \right)}} \log{\left(x \right)}} = 2$$
Solve:
$$x_{1} = \left(\frac{2^{\frac{3}{5}}}{2}\right)^{\sqrt{5}}$$
$$x_{1} = \left(\frac{2^{\frac{3}{5}}}{2}\right)^{\sqrt{5}}$$
This roots
$$x_{1} = \left(\frac{2^{\frac{3}{5}}}{2}\right)^{\sqrt{5}}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + \left(\frac{2^{\frac{3}{5}}}{2}\right)^{\sqrt{5}}$$
=
$$- \frac{1}{10} + \left(\frac{2^{\frac{3}{5}}}{2}\right)^{\sqrt{5}}$$
substitute to the expression
$$1 - \frac{\sqrt{1 - 4 \left(\frac{\log{\left(x \right)}}{\log{\left(4 \right)}}\right)^{2}}}{\frac{1}{\log{\left(4 \right)}} \log{\left(x \right)}} < 2$$
$$1 - \frac{\sqrt{1 - 4 \left(\frac{\log{\left(- \frac{1}{10} + \left(\frac{2^{\frac{3}{5}}}{2}\right)^{\sqrt{5}} \right)}}{\log{\left(4 \right)}}\right)^{2}}}{\frac{1}{\log{\left(4 \right)}} \log{\left(- \frac{1}{10} + \left(\frac{2^{\frac{3}{5}}}{2}\right)^{\sqrt{5}} \right)}} < 2$$
              ________________________________           
             /           /               ___\            
            /            |             \/ 5 |            
           /             |       / 3/5\     |            
          /             2|  1    |2   |     |            
         /         4*log |- -- + |----|     |            
        /                \  10   \ 2  /     /            
       /       1 - -------------------------- *log(4)    
      /                        2                      < 2
    \/                      log (4)                      
1 - -------------------------------------------------    
                    /               ___\                 
                    |             \/ 5 |                 
                    |       / 3/5\     |                 
                    |  1    |2   |     |                 
                 log|- -- + |----|     |                 
                    \  10   \ 2  /     /

Then
$$x < \left(\frac{2^{\frac{3}{5}}}{2}\right)^{\sqrt{5}}$$
no execute
the solution of our inequality is:
$$x > \left(\frac{2^{\frac{3}{5}}}{2}\right)^{\sqrt{5}}$$
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       x1
Solving inequality on a graph
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