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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).

David
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3 Answers3

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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.

Arthur
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  • I definitely agree that ylogx is unambiguous and one should write it that way. But I could use that same argument with the other form. log(xy) is unambiguous and so one should always write it that way if that is what they want. So, due to the ambiguity of log xy, should that be deemed meaningless just as the expression x - y - z) is technically meaningless without another opening parenthesis parenthesis since (x - y - z) does not equal x - (y - z)? – David Nov 01 '21 at 20:54
  • @David But $\log(xy)$ has additional symbols that $\log xy$ does not. On the other hand, $y\log x$ makes no such trade-off. Also, in $\log xy$, $x$ and $y$ are closer together than $\log$ and $x$, visually implying an order of operations. I think you will be hard-pressed to find someone who really thinks $\log xy$ means $(\log x)y$, while taking it to mean $\log (xy)$ is rather common. As for $x-y-z$, that's not ambiguous or meaningless at all: minuses are evaluated left-to-right along with additions. This is basically universally agreed on. – Arthur Nov 01 '21 at 22:32
  • I agree that x - y - z is not ambiguous. But x - y - z) is ambiguous as that value will depend upon where one inserts the open parenthesis. – David Nov 23 '21 at 04:37
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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.

William Ryman
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    One thing I found interesting about Desmos is that, while it does seem to treat multiplication as an operation to be performed because applying the logarithm, it does not extend that same courtesy to division. That is, if you type 1 = log xy, it will treat that as 1 = log (xy), while if you type 1=log x/y, it will immediately convert that to the equivalent of 1 = (log x)/y. – David Nov 01 '21 at 21:30
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If I encountered $\log x y$, I'd interpret it as $\log (x y)$.

If I were writing it, I'd write it as $\log (x y)$ or $y \log x$.

In the examination hall I would tend to use brackets rather than not, so as to extra clear for the examiner.

Prime Mover
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