Is there any reason that mean squared log error uses the natural log? Mean Squared Logarithmic Error uses the natural log. Why not use log base 2 or 10, for example?

By the change of base property of logarithms,
$$\log_a b = \frac{\log_c b}{\log_c a} = \frac{\ln b}{\ln a}, $$
so changing from any fixed base $a$ to any other base $c$ (or $e$) simply means dividing by a fixed value $\log_c a$, so a change of base is just a scaling. All logarithms are equivalent.
The natural logarithm to base $e$ is very common and indeed natural in many other statistical and mathematical settings, so it's often the first thing that comes to mind to statisticians (computer scientists might indeed rather think first of base $2$, or engineers to base $10$).
