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  1. If the equation $$\log_{10}(ax)\log_{10}(bx)+1=0$$ with $a>0$, $b>0$ constants has a solution $x>0$, it follows that $\frac ba \geq ans_{1}$ or $ans_{2}\geq \frac ba >ans_{3}$

  2. If $$\cos\theta=\sqrt{\frac 12+\frac1{2\sqrt2}}$$ and $$\sin\theta=-\sqrt{\frac 12-\frac 1{2\sqrt2}}$$ with $0\leq \theta <2\pi$, it follows that $2\theta=ans_{4}\pi$

  3. If $$y=\log_2\left(x+\sqrt{x_2+1}\right)$$ then $2^y-2^{-y}=ans_{5}x$

  4. The function $$f(x)=\log_2(\log_3(\log_2(\log_3(\log_2x))))$$ has the interval $x>ans_{6}$ as its maximum domain on real numbers

Find $ans_1,\cdots, ans_6$

I've no foundation in these but I'd appreciate if anyone could provide explanation for them. I've seen the answer but couldn't make anything from it.

EDIT: Sorry if this violated the rules here, apparently someone helped translate the whole thing into texts, I appreciate your help. Mods can you close the question? I'll do it right next time around :)

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closed as off-topic by JonMark Perry, pjs36, user99914, Zain Patel, user91500 May 14 '17 at 15:22

This question appears to be off-topic. The users who voted to close gave this specific reason:

  • "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – JonMark Perry, pjs36, Community, Zain Patel, user91500
If this question can be reworded to fit the rules in the help center, please edit the question.

  • $\begingroup$ Welcome to StackExchange - please type out your question using MathJax so that a) it isn't sideways and b) it is searchable on the site $\endgroup$ – lioness99a May 14 '17 at 11:59
  • $\begingroup$ For the last question, proceed from outside, and use the fact that $\log(1) = 0$ $\endgroup$ – samjoe May 14 '17 at 12:02
  • $\begingroup$ @lioness99a omg so embarassing, i'll change the question asap $\endgroup$ – cartmanbrahhhhh May 14 '17 at 12:07
  • $\begingroup$ For question $3$, use the fact that $$a^{\log_ab}=b$$ with $a=2$ $\endgroup$ – lioness99a May 14 '17 at 12:15
  • $\begingroup$ Welcome. It looks like your question is on the brink of being closed. This is probably because it's been perceived as a do-my-homework-for-me task, which is not what this site is for. Dumping a handful of questions at once doesn't help. Generally, you should provide some sense of what you know and don't know about the problem. (You say you've seen the answers; include them for completeness.) This helps us tailor answers to you, and prevents us from telling you things you already know. Remember: You are asking a favor of the people here; try to make answering your question as easy as possible. $\endgroup$ – Blue May 14 '17 at 12:51
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NOTE: I removed the answer due to Blue's comment above. If you already have it, fine, but refer to Blue's comment for asking future questions.

Hints for #2:

  • Square $\cos \theta$ and $\sin \theta$.

  • Since we're looking for $2 \theta$, which trigonometric function applies?

  • Your angles within the range you're looking for are in the first and fourth quadrants.

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