What is base of $\log$ in Subexponential Algo for DLP?
In this case, the base does not matter as the $\log$ terms are wrapped in an $O$ expression. The $O$ expression lets you throw away constant factors and to convert something base $X$ to base $Y$ is simply $\log_X Z = \frac{\log_Y Z}{\log_Y X}$, well, $\log_Y X$ is a constant, so it can be thrown out in the $O$ expression.

The reason the second reference explicitly use $\ln$ is that the $O$ expression includes an exponent that does not also contain an $O$ expression. In this case, the base does matter.