What is the base of $\log x$? I've seen "$\log x$" being used in some papers (and by Wolfram|Alpha), and I was confused because so far I have only ever seen the $\log$ used with a base ( so e.g. $\log_y x$).
Am I correct that $\log x = \log_e x = \ln x$?


*

*If so, why is $\log x$ used over $\ln x$? Isn't the letter more expressive
and less confusing?

*If not, what is the base of $\log x$?

This depends heavily on context, but the most common definition is $\log x=\ln x$. That said, in introductory textbooks for students, $\log$ is often used to mean $\log10$, and in coding theory $\log$ often refers to $\log2$. In general though, unless specified it typically refers to $\log_e$. 
As to why there's no standardised convention it's hard to say for certain, but most likely just historical reasons: authors picked convention to what was most convenient, and that 'stuck' within each field of mathematics. More details about this could be found here.
