Let $x \in \mathbb{N}$.
For $\infty>x>e$,
Let $x$ be the largest prime in the factorization of the primorial $x\#$.
How does one go about proving $x*\ln(\ln(x))>\ln(\ln(x\#))$?
The plot of LHS$-$RHS looks like it will be always positive, so heuristically it seems that there should be a simple proof, but so far it has eluded me. Here is my attempt:
From the definition of the primorial, we easily have
$n*\ln(x)>\ln(x\#)$
where $n$ is the number of primes in $x$. Then,
$\ln(n)+\ln(\ln(x))>\ln(\ln(x\#))$
$x*\ln(\ln(x))>x*(\ln(\ln(x\#))-\ln(n))>\ln(\ln(x\#))-\ln(n)$
So, at least it's true for sufficiently small $n$.
$\ln(\ln y)$ is unbounded, that is, it can be made as large as you like for a large enough $y$. Without any other constraint on $x$ and $y$, the problem seems incomplete