Is "$\log xy$" ambiguous?

How should one interpret "$\log xy$"? Choices are

1. it is the same as $\log(xy)$, for which there is no ambiguity
2. without parentheses, only the $x$ is in the logarithm, and so it is equivalent to $\log(x)y$, which most would write as $y\log(x)$, for which there is no ambiguity
3. it is not well defined, and so it has no meaning

People may have an opinion on this question, but if you can cite your answer from a reputable website or a book, I would greatly appreciate it as I have had no such luck.

Note that this ambiguity does not occur when we are taking the log of a fraction as the log is written level with the division line and so the entire fraction can be viewed as one object.

Update: one reason I ask this question is that in grading student's work, I need to decide whether or not to count off for incorrect notation if they write logxy when they mean log(xy).

Solution 1:

If someone meant $y\log x$, they would write that in order to avoid any ambiguity. Thus it is reasonable to assume $\log xy$ means $\log(xy)$.

But many prefer not to assume at all, as it must be done case-by-case without rigorous rules. And if you have to assume, you can never be certain what the author meant, at least without context.

Solution 2:

I don't know if the Desmos graphing calculator counts as an authority on the subject, but it seems to interpret it as option 1. Most people that I know, including myself, would also interpret it as option 1. This also seems to be the case for other such parentheses-less functions, such as $\sin xy$. If any ambiguity is caused, it's good practice to put parentheses around the argument of a function for clarification.