Doubt in raising a power to a complex number

Question

What's the value of $$i^{i^{i^{...}}}$$?

I tried to take log on both sides.

$x=i^x$

$\implies \log x=x \log i$

After this how can I solve this... I am sorry, that I don't know the methods you are explaining... And more over that the question is in school level only..

Answer 1

Such number $x$ statistifies $i^x=x$. This can be solved with Lambert W function. Doing so, gives:

$$x \approx -1.861743075013160441612498412-0.4107999688363923527986399873 i$$ $$x \approx 0.4382829367270321593816295045+0.3605924718713854895879688663 i$$

But we see that they are always positive, so it is the second one.