Does implicit multiplication take precedence over functions? Does implicit multiplication take precedence over functions? For example, what is meant by $\sin 2(x+2)$, $2x!$, $\log 2x$, etc.?
 A: Though this is a fairly elementary question, I think it deserves a proper answer.
How things should be done:
Things like $\sin2x$ and $\log y^2$ are technically an abuse of notation. Unless it's very obvious from context, you should use brackets, and instead write $\sin(2x)$ and $\log(y^2)$ to make sure there is no ambiguity (precedence is made clear by the brackets). The only common function that doesn't usually use brackets in its notation is the factorial, and even then, if you ever need to write the factorial of $2n$, you should write $(2n)!$ with brackets.
In other words, there is usualy no need for some special precedence rule for functions, as the standard precedence is inherited from brackets. When authoring math, your job is to clearly convey the meaning of your work to the reader, and if your notation is ambiguous to a reader, you have failed that reader.
The reality of things:
Sometimes people in disciplines like mathematics, engineering, and physics where math is very frequently used will take shortcuts in their writing, often abusing notation to save on space and on effort when writing. For example, as David K mentions in the comments, it is not uncommon to find things like $\cos 2x=2\sin x\cos x$ or $\log xy=\log x+\log y$ in literature and teaching environments. Usually (one would hope) that this is only done when meaning is clear from context and convention, but if you're not already familiar with common conventions, things like this might be hard to parse. To help you, here is a small list of common conventions:

*

*Addition is usually performed after applying the function. For example, $\log x+y$ usually means $\log(x)+y$ rather than $\log(x+y)$.


*Multiplication by a constant (or variable) is usually performed before applying the function. For example, $\tan ax$ usually means $\tan(ax)$ rather than $\tan(a)x$, and $\log 2(x+1)$ usually means $\log(2(x+1))$ rather than $\log(2)(x+1)$. This is especially true for trig functions, which one very often encounters evaluated at constant multiples of a variable. The one exception is the factorial, where $cn!$ always means $c(n!)$.


*When two (or more) functions are involved, multiplication is usually performed after application of the functions. For example, $\cos x\sin y$ usually means $\cos(x)\sin(y)$ rather than $\cos(x\sin(y))$.


*Exponentiation and division is usually performed before applying the function. For  example, $\arctan x/y$ usually means $\arctan(x/y)$ rather than $\frac{\arctan(x)}{y}$, and $\sin x^2$ usually means $\sin(x^2)$ rather than $(\sin(x))^2$. Authors who mean the latter, will instead usually write $\sin^2 x$ or $(\sin x)^2$. Again, the exception is the factorial, where $m/n!$ should always mean $\frac{m}{n!}$.
Remember that you shouldn't assume these conventions are universal. It's always a good idea to look for context, and appeal to common sense when interpreting written math. If you're unsure, you can often even plug in numbers to check if a particular interpretation of an ambiguous formula is the intended one.
A: Brief answer. Although it is a mathematical abuse omitting brackets, let us assume that something as $\sin2(x+2)$ has been found on a little piece of paper (of a paper, maybe... just to be fun) and we have no clue from the context.
Now, my natural interpretation would be to read $\sin2(x+2)$ as $\sin(2(x+2))$, since I consider as an argument of the rightmost function to the left everything that preceds any operator to the right (or the end of the expression itself).
The interpretation can be different if I already know the style of the author (e.g., a math professor who usually hastly omits brackets during his speech/lesson or if there is some clarification at the beginning of the manuscript/paper we are reading).
Generally speaking, I give the highest priority to everything that follows any mathematical function up to a summation or another function (reading from left to right as usual), since we can write $ab$ meaning $a \cdot b$ and we cannot interpret it as $a + b$, taking the above as the argument of the function itself.
Thus, having no clue from the context, my final answer is to interpret "$2 \sin 2(x+2)$, $fact2x$, $\log2x$" as "$2 \cdot \sin(2\cdot(x+2))$, $(2 \cdot x)!$, $\log(2 \cdot x)$" (respectively).
