Asymptotics of $f(z)$ where $z=\int_2^{f(z)} \frac{dx}{\ln(x)}$

Question

I am trying to determine what the behavior of the inverse logarithmic integral is as $z\to \infty$. I noticed that $f'(z)=\ln(f(z))$ which follows from differentiating $$z=\int_2^{f(z)} \frac{dx}{\ln(x)}$$ but I do not know if that could help in this situation. If anyone has any ideas or methods on how to find the asymptotics of this function it would be greatly appreciated.

Answer 1

Since $\int_2^{z} \frac{dx}{\ln(x)} $ is essentially the logarithmic integral (see https://en.wikipedia.org/wiki/Logarithmic_integral_function), what you are looking for is the inverse of this.

A search for "inverse of logarithmic integral" comes up with a number of good hits including this, here: Inverse logarithmic integral