Find a real value $y$ for that the identity holds Question:  Find a real value $y$ such that the following identity holds
$$\log(\log(\log(x)))=\int^x_y\frac{dx}{x\log(x)\log(\log(x))}\quad (x>y)$$
Then, find an algorithm to calculate
$$f_n(x)\quad (x>e^{n-1})$$
where $f_0(x)=x$ and $f_n(x)=\log(f_{n-1}(x))$.
Thanks in advanced!
Help appreciated!!

Make the substitution $u=\log(\log x)$, We find that one antiderivative is $\log(|\log(\log x)|)$. 
Substitute the two endpoints. So we want $\log(\log(\log y))=0$. That means $\log(\log y)=1$, so $\log y=e$, and therefore $y=e^e$. 
Note that if we want the iterated logarithm to be defined, in general $x\gt e^{n-1}$ will not be enough. We need an iterated exponential, as in the calculation of $y$.  
