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human
CMV: “Log x” should never be used to mean “log base e of x/ln x”

In calculus and higher math classes, there is an epidemic of professors and mathematicians that casually use “log x” when they really mean “ln x.” This behavior is not only nonintuitive, but is also confusing, to the detriment of student learning.

- from the moment students are taught about logarithms, they are taught that “log x” means “log base 10 x.” This implication is used in science, math, and calculator buttons. It’s confusing when professors in higher math classes start using it to mean something else.

- there is ALREADY a notation for “natural log x”: it’s ln x. I get that natural logs are easier to work with in calculus, but it’s silly to redefine what “log x” means when ln x is already defined to mean this.

- “ln x” is not only already defined, but it is actually less work to write it. It takes one less chalk, key, or pencil stroke to write “ln x” than “log x.” So even from a pure time/efficiency point of view, it’s better to use “ln x” than “log x.”

In conclusion, there is no good reason besides convention to say “log x” when you mean “ln x,” no matter what level of math you are doing. For the benefit of students everywhere, this irrational behavior should be recognized for what it is stamped out.

Ok, my view has changed: I can see that mathematicians value log x as base e, whereas engineers argue for log x base 10. I don’t believe using log x as ln x needs to be stamped out, but it might be worth looking at how we teach logarithms in the first place.
gpt
> it’s silly to redefine what “log x” means when ln x is already defined to mean this.

You only think it is *re-defined* because of your personal learning arch.  Log has a long history of having a base of *e*.

If you want to specify base 10 unambiguously, you can either use log_10 or lg.

> It’s confusing when professors in higher math classes start using it to mean something else.

Unfortunately sometimes notation is confusing, but the goal of notation isn't to make it simple for new learners, but rather for people in the field to communicate their ideas.  And in order to join that field, you must be trained in the notations that the people in the field use to not only understand current notation, but historical notation.  If they avoided teaching that log can sometimes mean base e in many fields, you'd have trouble understanding the published work from the past in that field.

Many fields have a standard that log means base e, and if it isn't clear it becomes clear the first time someone uses log in a paper.  I've never seen someone jump back and forth between the two without doing something to more explicitly establish what base it is.