Mister Exam
log(5)^2*(25-x^2)-3*log(25-x^2)+2>=0 inequation
The solution
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2 / 2\ / 2\ log (5)*\25 - x / - 3*log\25 - x / + 2 >= 0
$$\left(\left(25 - x^{2}\right) \log{\left(5 \right)}^{2} - 3 \log{\left(25 - x^{2} \right)}\right) + 2 \geq 0$$
(25 - x^2)*log(5)^2 - 3*log(25 - x^2) + 2 >= 0
Detail solution
Given the inequality:
$$\left(\left(25 - x^{2}\right) \log{\left(5 \right)}^{2} - 3 \log{\left(25 - x^{2} \right)}\right) + 2 \geq 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\left(\left(25 - x^{2}\right) \log{\left(5 \right)}^{2} - 3 \log{\left(25 - x^{2} \right)}\right) + 2 = 0$$
Solve:
$$x_{1} = - \frac{\sqrt{25 \log{\left(5 \right)}^{2} + 3 W\left(- \frac{e^{\frac{2}{3}} \log{\left(5 \right)}^{2}}{3}\right)}}{\log{\left(5 \right)}}$$
$$x_{2} = \frac{\sqrt{25 \log{\left(5 \right)}^{2} + 3 W\left(- \frac{e^{\frac{2}{3}} \log{\left(5 \right)}^{2}}{3}\right)}}{\log{\left(5 \right)}}$$
Exclude the complex solutions:
This equation has no roots,
this inequality is executed for any x value or has no solutions
check it
subtitute random point x, for example
$$2 + \left(- 3 \log{\left(25 - 0^{2} \right)} + \left(25 - 0^{2}\right) \log{\left(5 \right)}^{2}\right) \geq 0$$
so the inequality is always executed
$$\left(\left(25 - x^{2}\right) \log{\left(5 \right)}^{2} - 3 \log{\left(25 - x^{2} \right)}\right) + 2 \geq 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\left(\left(25 - x^{2}\right) \log{\left(5 \right)}^{2} - 3 \log{\left(25 - x^{2} \right)}\right) + 2 = 0$$
Solve:
$$x_{1} = - \frac{\sqrt{25 \log{\left(5 \right)}^{2} + 3 W\left(- \frac{e^{\frac{2}{3}} \log{\left(5 \right)}^{2}}{3}\right)}}{\log{\left(5 \right)}}$$
$$x_{2} = \frac{\sqrt{25 \log{\left(5 \right)}^{2} + 3 W\left(- \frac{e^{\frac{2}{3}} \log{\left(5 \right)}^{2}}{3}\right)}}{\log{\left(5 \right)}}$$
Exclude the complex solutions:
This equation has no roots,
this inequality is executed for any x value or has no solutions
check it
subtitute random point x, for example
x0 = 0
$$2 + \left(- 3 \log{\left(25 - 0^{2} \right)} + \left(25 - 0^{2}\right) \log{\left(5 \right)}^{2}\right) \geq 0$$
2
2 - 3*log(25) + 25*log (5) >= 0
so the inequality is always executed