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lg^2(x)+2lg(10)(x)>3 inequation

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   2                     
log (x) + 2*log(10)*x > 3

x 2 \log{\left(10 \right)} + \log{\left(x \right)}^{2} > 3

x*(2*log(10)) + log(x)^2 > 3

Detail solution

Given the inequality:

x 2 \log{\left(10 \right)} + \log{\left(x \right)}^{2} > 3

To solve this inequality, we must first solve the corresponding equation:

x 2 \log{\left(10 \right)} + \log{\left(x \right)}^{2} = 3

Solve:

x_{1} = 0.59161255434212

x_{2} = 0.262102533701829

x_{1} = 0.59161255434212

x_{2} = 0.262102533701829

This roots

x_{2} = 0.262102533701829

x_{1} = 0.59161255434212

is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:

x_{0} < x_{2}

For example, let's take the point

x_{0} = x_{2} - \frac{1}{10}

- \frac{1}{10} + 0.262102533701829

0.162102533701829

substitute to the expression

x 2 \log{\left(10 \right)} + \log{\left(x \right)}^{2} > 3

0.162102533701829 \cdot 2 \log{\left(10 \right)} + \log{\left(0.162102533701829 \right)}^{2} > 3

3.31067566481757 + 0.324205067403657*log(10) > 3

one of the solutions of our inequality is:

x < 0.262102533701829

 _____           _____          
      \         /
-------ο-------ο-------
       x2      x1

Other solutions will get with the changeover to the next point
etc.
The answer:

x < 0.262102533701829

x > 0.59161255434212

Solving inequality on a graph