$ Π (x) ge log log x $ holds $ 2 le x e e {{e ^ 3} <5.3 times 10 ^ 8 $?

The book Theory of Numbers by G H Hardy, et al. test $ π (x) ge log log x $ for $ x> e ^ {e ^ 3} $. There is a way to try out that also goes for $ 2 le x e e ^ {e ^ 3} $ otherwise (in the worst case) some valuable source to verify this numerically?