Meaning of $\log$ If you write $\log{x}$ rather than ${\log_a{x}}$ for some base $a$, does it have a particular meaning? Sometimes I see people leave off the base by mistake when posting questions and it seems from the comments they get that it doesn't simply mean "log of $x$ to some base but I haven't said which yet".  For example, does it indicate the complex logarithm?
 A: In mathematics, this usually means log in base $e$.
In computer science, this sometimes means log in base $2$.
In natural sciences (such as chemistry) and high school, this usually means log in base $10$.
Finally, when it appears in big-O notation (such as $O(\log x)$), the base doesn't matter, since the difference is a multiplicative constant.
Edit (answering your edit): The complex logarithm is a different matter than bases. The complex logarithm is always taken in base $e$ (it doesn't make much sense otherwise), and the principal branch is usually denoted $\text{Log}$ with a capital L. Usually it is pretty clear from the context whether a complex logarithm is being discussed, since it must be handled with care, with its branch cut or with its special domain as a multi-valued function.
A: In many contexts the base doesn't make much, or any, relevant difference and thus can safely be omitted. When it does make a difference, context can determine the base (e.g., when talking about bit complexity, base $2$ makes most sense). In most mathematical contexts, the base, if omitted, is assumed to be $e$. 
