In my calculus classes, I prefer to use log(x) to mean log[sub]e/sub because of one such “interesting property”, namely the derivative. The derivative of log[sub]e/sub is just 1/x, while the derivative of log[sub]10/sub is log[sub]10/sub/x. Rather than try to remember that ugly-looking constant every time, I just use e as the base of my logarithms by default. There’s a similar justification for using radians to measure angles: using radians the derivative of sin(x) is cos(x), but using degrees the derivative of sin(x) is cos(x)*pi/180. Ugly-looking constants and bad and should be set to 1 whenever it’s reasonable to do so, IMO.
That is exactly the interesting property I was thinking of - but I didn’t want to bring up calculus for someone who was having trouble with logs. And come to think of it, log base 10 doesn’t seem to be used much past pre-calculus, especially now that we don’t have humungo slide rules in the front of high school math classes.
As for me, things started breaking down when we got to imaginary exponents. I think it’s genetic - Feynman’s description of how he got calculus intuitively is profoundly disheartening.