Should I use parentheses when writing $\log$? Should I be using parentheses when using things like $\log$ in LaTeX, and when handwriting?
Should I use $\log x$ or $\log(x)$?
If it's just one value or variable, I can see getting away with not using parentheses, but suppose I have a function in there:
$$
    \log p(w|v) \; \text{or} \; \log( p(w|v) ) \; ?
$$
Same question applies to things like $\sin$, $\arctan$, $\ln$, etc.
 A: If $f$ was a function and $x$ its argument: who in the mathematician's world would write $f\,x$ instead of $f(x)$? So the question really is: “What is the reason to omit the argument's brackets in some cases?”  (And how to remember these exceptions?  It's sort of remembering the list of all irregular verbs.) Besides of overcame  traditional unprogressive  reasons every teacher among us will agree that maximizing the data-ink ratio will as well minimize the readability, hence the grasping of what is written. 
In case you're a teacher you should always put brackets around a function's argument, just for the sake of consistency.  In case you're not: correct your habit to avoid ambiguous expressions: Don't spare (digital) ink, write out your thoughts  as clearly as possible!  
A: I treat $\log$ similarly to the trigonometric functions, so I would write $\sin x$ but I would write $\sin(x+k\pi)$ when the expression would be ambiguous without the parentheses.
So similarly, $\log x$, $\log f(x)$ and $\log\dfrac{1-x}{x}$ but $\log (1-x)$ and $\log(f(x)+g(x))$.
A: See the answer here: https://math.stackexchange.com/a/212201/88378
If it's not ambiguous, omitting parentheses can help readability. 
$\log$ and $\sin$ are different from $f$ and $g$. The latter could denote a scalar for example, and putting $()$ after them makes it clear that they're functions. $\log$, $\sin$, etc are universally accepted to denote specific functions, so parentheses aren't necessary to clarify that.
