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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

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closed as off-topic by YuiTo Cheng, Claude Leibovici, Nosrati, John Omielan, zipirovich Jun 9 at 4:49

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  • $\begingroup$ Welcome to Mathematics Stack Exchange! A quick tour will enhance your experience. Here are helpful tips to write a good question and write a good answer. For equations, use MathJax. $\endgroup$ – dantopa Jun 9 at 3:30
  • $\begingroup$ Why was my very first question of this site is being voted -1 and closed.what is math jax and how to use it.please give me a answer so that i can use it in my next questions. $\endgroup$ – Aryan 24k Jun 9 at 8:49
  • $\begingroup$ As noted in the shaded box, the question was missing context and details of your efforts. The site discourages dumping homework questions, and in this case five different users voted. A benevolent user has converted your formulation into MathJax, and this example should get you going. Please checkout the web page for writing questions, and keep posting. $\endgroup$ – dantopa Jun 9 at 13:25
  • $\begingroup$ I have tried the question but nothing useful information was pulled out.so i asked the question as it is.also i didn't know about math jax so i posted it as i can.so it's ok... $\endgroup$ – Aryan 24k Jun 9 at 14:15
  • $\begingroup$ Consider adding a sentence or two about context. Where did this problem arise? Where are you in your studies? Also, demonstrate some effort and some familiarity with the material. Show your familiarity with exponential and logarithmic functions. And use the search function to look at similar questions. Good luck. $\endgroup$ – dantopa Jun 10 at 2:50
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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 $\displaystyle y^2 -3y+ 2 > 0$.

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  • $\begingroup$ Thanks for your help.. $\endgroup$ – Aryan 24k Jun 10 at 4:29

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