Time complexity to solve Discrete log problem What is the time to complexity to solve
Discrete log  problem now in $Z_p$?

Initially for $n$ bit prime $p$, it was $Exp(n^{1/3})$.
These days the fastest general method to solve discrete logarithms modulo primes is the number field sieve, which has the asymptotic complexity 

$$
e^{(1.92+o(1)) (\log p)^{1/3} (\log\log p)^{2/3}}
$$