Solve the following Algebraic Logarithmic inequalities

Question

Solve the following logarithmic inequality with all log base $10$. $$\left(\frac12\right)^{(\log x^2)} + 2 > 3\times2^{(-\log(-x))}$$ I have done many logarithmic inequalities but in this i am not able to crack the problem.please give the hint or the approach on should try while doing these type of question.i am preparing for international mathematics olympiad so any help would be appreciate. Thanks

Answer 1

Hint:

We need $-x>0$.

Let $\displaystyle y=2^{-\log(-x)}$.

Then $\displaystyle \left(\frac12\right)^{\log x^2}=\left(2^{-\log(-x)}\right)^2=y^2$.

The inequality can be written as $\displaystyle y^2 -3y+ 2 > 0$.