Mister Exam

Other calculators

Solving inequalities/ lg^2x-4lgx-5>0

lg^2x-4lgx-5>0 inequation

A inequation with variable

The solution

You have entered [src] 
   2                      
log (x) - 4*log(x) - 5 > 0
(log⁡(x)2−4log⁡(x))−5>0\left(\log{\left(x \right)}^{2} - 4 \log{\left(x \right)}\right) - 5 > 0 
log(x)^2 - 4*log(x) - 5 > 0
Detail solution
Given the inequality:
(log⁡(x)2−4log⁡(x))−5>0\left(\log{\left(x \right)}^{2} - 4 \log{\left(x \right)}\right) - 5 > 0
To solve this inequality, we must first solve the corresponding equation:
(log⁡(x)2−4log⁡(x))−5=0\left(\log{\left(x \right)}^{2} - 4 \log{\left(x \right)}\right) - 5 = 0
Solve:
x1=e−1x_{1} = e^{-1}
x2=e5x_{2} = e^{5}
x1=e−1x_{1} = e^{-1}
x2=e5x_{2} = e^{5}
This roots
x1=e−1x_{1} = e^{-1}
x2=e5x_{2} = e^{5}
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
x0<x1x_{0} < x_{1}
For example, let's take the point
x0=x1−110x_{0} = x_{1} - \frac{1}{10}
=
−110+e−1- \frac{1}{10} + e^{-1}
=
−110+e−1- \frac{1}{10} + e^{-1}
substitute to the expression
(log⁡(x)2−4log⁡(x))−5>0\left(\log{\left(x \right)}^{2} - 4 \log{\left(x \right)}\right) - 5 > 0
−5+(log⁡(−110+e−1)2−4log⁡(−110+e−1))>0-5 + \left(\log{\left(- \frac{1}{10} + e^{-1} \right)}^{2} - 4 \log{\left(- \frac{1}{10} + e^{-1} \right)}\right) > 0
        2/  1     -1\        /  1     -1\    
-5 + log |- -- + e  | - 4*log|- -- + e  | > 0
         \  10      /        \  10      /

one of the solutions of our inequality is:
x<e−1x < e^{-1}
 _____           _____          
      \         /
-------ο-------ο-------
       x1      x2

Other solutions will get with the changeover to the next point
etc.
The answer:
x<e−1x < e^{-1}
x>e5x > e^{5}
Solving inequality on a graph
-5.0-4.0-3.0-2.0-1.05.00.01.02.03.04.0-200200
Rapid solution 2 [src] 
     -1      5     
(0, e  ) U (e , oo)
x in (0,e−1)∪(e5,∞)x\ in\ \left(0, e^{-1}\right) \cup \left(e^{5}, \infty\right) 
x in Union(Interval.open(0, exp(-1)), Interval.open(exp(5), oo))
Rapid solution [src] 
  /   /            -1\     /         5    \\
Or\And\0 < x, x < e  /, And\x < oo, e  < x//
(0<x∧x<e−1)∨(x<∞∧e5<x)\left(0 < x \wedge x < e^{-1}\right) \vee \left(x < \infty \wedge e^{5} < x\right) 
((0 < x)∧(x < exp(-1)))∨((x < oo)∧(exp(5) < x))
    © Mister Exam – Calculator