1
$\begingroup$

Very based question regarding expression transformation.

$$\log p_i \leq \log i + 2 \log\log i $$ $$ p_i \leq i(\log i)^2$$

I am not getting this transformation. So from what I try:

$$\log p_i \leq \log (2i \log\log i) $$ $$ p_i \leq 2i \log\log i$$ $$ p_i \leq 2i \log^2 i$$ $$ p_i \leq \log^2 i^{2i}$$

And I am stuck...

$\endgroup$
2
$\begingroup$

We have $$log(p_i)\leq log(i)+2\log(log(i))$$ $$log(p_i)\leq log(i)+\log(log(i)^2)$$ $$log(p_i)\leq \log(i\log(i)^2)$$ $$p_i\leq i\log(i)^2$$

Here the second inequality follows since $a\log(b)=\log(b^a)$, and the third since $\log(a)+\log(b)=\log(ab)$. The last inequality holds since th logarithm is an increasing function.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.