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lg^(2)(x)=lg(10x) equation

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Numerical solution:

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The solution

You have entered [src]
   2               
log (x) = log(10*x)
$$\log{\left(x \right)}^{2} = \log{\left(10 x \right)}$$
The graph
Sum and product of roots [src]
sum
       _______________          _______________
 1   \/ 1 + 4*log(10)     1   \/ 1 + 4*log(10) 
 - - -----------------    - + -----------------
 2           2            2           2        
e                      + e                     
$$e^{\frac{1}{2} - \frac{\sqrt{1 + 4 \log{\left(10 \right)}}}{2}} + e^{\frac{1}{2} + \frac{\sqrt{1 + 4 \log{\left(10 \right)}}}{2}}$$
=
       _______________          _______________
 1   \/ 1 + 4*log(10)     1   \/ 1 + 4*log(10) 
 - + -----------------    - - -----------------
 2           2            2           2        
e                      + e                     
$$e^{\frac{1}{2} - \frac{\sqrt{1 + 4 \log{\left(10 \right)}}}{2}} + e^{\frac{1}{2} + \frac{\sqrt{1 + 4 \log{\left(10 \right)}}}{2}}$$
product
       _______________        _______________
 1   \/ 1 + 4*log(10)   1   \/ 1 + 4*log(10) 
 - - -----------------  - + -----------------
 2           2          2           2        
e                     *e                     
$$\frac{e^{\frac{1}{2} + \frac{\sqrt{1 + 4 \log{\left(10 \right)}}}{2}}}{e^{- \frac{1}{2} + \frac{\sqrt{1 + 4 \log{\left(10 \right)}}}{2}}}$$
=
E
$$e$$
E
Rapid solution [src]
            _______________
      1   \/ 1 + 4*log(10) 
      - - -----------------
      2           2        
x1 = e                     
$$x_{1} = e^{\frac{1}{2} - \frac{\sqrt{1 + 4 \log{\left(10 \right)}}}{2}}$$
            _______________
      1   \/ 1 + 4*log(10) 
      - + -----------------
      2           2        
x2 = e                     
$$x_{2} = e^{\frac{1}{2} + \frac{\sqrt{1 + 4 \log{\left(10 \right)}}}{2}}$$
x2 = exp(1/2 + sqrt(1 + 4*log(10))/2)
Numerical answer [src]
x1 = 0.333643853685936
x2 = 8.14725581912804
x2 = 8.14725581912804