The asymptotic running time of the best discrete log algorithm is
approximately the same as for the best general purpose factoring
algorithm. Therefore, it requires about as much effort to solve the
discrete log problem modulo a 512-bit prime as to factor a 512-bit RSA
modulus. There are experimental evidences that the discrete log problem
is slightly harder than factoring: it is suggested that the effort
necessary to factor a 110-digit integer is the same as the effort to
solve discrete logarithms modulo a 100-digit prime. This difference is
so slight that it should not be a significant consideration when
choosing a cryptosystem.
Historically, it has been the case that an algorithmic advance in either problem, factoring or discrete logs, was then applied to the other. This suggests that the degrees of difficulty of both problems are closely linked, and that any breakthrough, either positive or negative, will affect both problems equally.