TITLE: Where is the fallacy in the argument using Prime Number Theorem
QUESTION [10 upvotes]: I am reading about Prime Number Theorem from book by Ingham. As as application of PNT I found the following theorem:

Now my doubt is at the step $\frac{\log(y)}{\log(x)}\rightarrow 1$, we can say $\log(y)\rightarrow\log(x)$ and if I apply antilog I get $y\rightarrow x$ which is not true by PNT.
So where is the fallacy in my argument? 
I think it is because we are neglecting $\log\log x$ term, but I am not sure and I am unable to find a suitable argument.
Also we are applying $\log$ on PNT even though it is kind of limit, So why cant we apply antilog. Are there any conditions to apply some function for these kind of cases?

REPLY [2 votes]: Here is where little $o$ and big $O$ notation can clarify things greatly.  You have that $$\log y=(1+o(1))\log x.$$  Exponentiating, this implies that $$y=x^{1+o(1)}.$$  The important thing is that the function $x^{1+o(1)}$ does not necessarily converge to $1$ - in fact it can be quite large.  Let $f(x)=\frac{K\log \log x}{ \log x}$ is $o(1)$, and in this case $$x^{f(x)}=\exp\left(\frac{K\log \log x}{ \log x}\cdot \log x\right)$$ $$=(\log x)^K.$$