I think stereotypically that log means base 10 and ln means base e. That is how all the 8 different high school and college texts (published 30s to 80s) that I have use it. But I have noticed a (maybe growing?) tendency for other usage. Had to get a formal paper in econ to define their base. (And they were using the log means base e usage, that is more rare...at least for knuckle draggers like me.)
I did a quick Google search on does log mean log base 10 and their were several university math tutor sites still propounding the orthodox religion about common logs. But, a few heretics were seen also. :-(
"So, when you see log by itself, it means base ten log. When you see ln, it means natural logarithm (we'll define natural logarithms below). In this course only base ten and natural logarithms will be used."
U Minn: http://www.mclph.umn.edu/mathrefresh/logs.html#:~:text=So%2C%20when%20you%20see%20log,natural%20logarithms%20will%20be%20used.
My advice would be to stick with the log means 10 usage since it is still (I think) the most common usage and it's your class that they mostly need it in. I still remember fondly log tables (base 10) being handed out for AP chem into the 90s, in case kids did not have a calculator. ;-)
Maybe briefly mention it to the kids, that some usage is different. (heck "billion" means something different in different countries...and don't get me started on the comma/decimal hassles when passing documents back and forth with Germans).
I wouldn't feel too worried if they see something different in the future. That can be dealt with then by the other teacher. Or if they are really doing sophisticated work in econ or pure math, than they'll be quite capable of dealing with the momentary notation confusion.
Also, FWIW, I wouldn't worry about alternate usage in the more rarified air of higher math courses. Most of your kids need to learn things the old school way and they will be heading for the natural sciences or engineering (which are more traditional). If they end up reading a measure theory text, there's other parts of the text that will vex them more than Napierian being treated as common. ;)
P.s. Of course if you're tutoring and you learn the teach has other usage, defer to him. Or her. Or it.