Using $\ln$ or $\log$

For any applicable base $b$ you have $$\log_b x = \frac 1{\ln b}\cdot\ln x$$ so $O(\log_b f(n))$ is exactly the same as $O(\ln f(n))$.

Sometimes $\log$ is used as a 'general' logarithm with the base unspecified, although constant and greater than $1$ (which is useful when multiplicative constant does not matter), while $\ln$ is log with base $e$ and $\lg$ is a logarithm with base $10$.
Sometimes $\log$ has a default base $2$, for example in computer science.

I suggest you use $\ln$ in the paper and $\log$ when you have order results (since the base of $\log$ does not matter, but it is conventional to use $\log$ when stating order results).