I have a contour plot with automatically generated numeric labels on the contours themselves, and question how the subtitle identifying the labels should read. The number represents log10(time) where time is in seconds. In other graphs I consistently label the meaning of the contour numbers with phrases like “Board thickness, cm”, and I want this plot to be consistent. Should I say, in this case, “Time, log10(s)”? This makes s look like a variable to me. Should I say “log10(Time), s”? This makes it look like seconds is the unit of the logarithm itself, which isn’t right. Should I say “log10(Time, s)”? This looks funny, like the log10 function takes two arguments. What’s standard practice?
This sounds like something your local style guide would handle, but if you don’t have one… how about “Time, log[sub]10/sub, s”? I also have in the back of my mind car tachometers, where they say “rpm x 1000”. Maybe simply “log[sub]10/sub, s”? The base-ten logarithm of the time would still be in seconds, wouldn’t it?
I would go with log[sub]10[/sub][Time (s)]. Or, if you want to avoid the whole thing, just label the axes with 10[sup]0[/sup], 10[sup]1[/sup], 10[sup]2[/sup], etc. rather than 0, 1, 2, etc.
I think I like this option particularly because the phrase “Time (s)” resolves to a dimensionless number. At least, I see it that way.
This isn’t on the axes, it’s referring to labels embedded in the contours themselves. My tool doesn’t support changing them, other than specifying their size. I have a subtitle at the top with the note I’m describing in it, to describe the “third axis” variable.
No. A logarithm can’t have any units, and it’s ill-defined for the argument of a logarithm to have units, either.
By saying subtitle on the map, do you mean the key?
If so, I’d say something like “Contours are log10 Time, in seconds.”
~VOW
Yeah, I was unsure of that. So it would be, “the logarithm of (the time in seconds)”, rather than “(the logarithm of the time) in seconds”.
The important part is to make it clear that the number is in fact a log.
That done, Time, log10(sec) is the simplest and clearest to me.
I like this. It seems clear and does not seem to specifically say that the log has units.
This is a graph, not a map. I am putting the note on the second line at the top, under the title. I am using SAS, which considers the main title “title1” and the second line “title2”. I think of the key as text at the bottom that identifies each contour individually in a list (SAS calls this a legend) and my note is making a statement about all contours taken together. The contour values actually appear someplace on the contours themselves, which break around the numbers.
Napier:
Sorry for my confusion. I worked with maps for a hundred years, and my brain seem to only function in that reference.
But your description of the way your graph is labeled sounds like common map usage. The contour is essentially a closed circle, with a small break where the number is printed. Also every fifth contour, or every tenth, is usually a heavier line, to make the interpretation just a tad easier.
~VOW
Can you re-plot them in units of 10 * log10(time) instead? Then the label could be “dB seconds” or just “dB s”, which should be familiar to scientist and engineers.
Oh, ZenBeam, I LOVE you! Well, wait, calm down - still, I tremendously appreciate your post!
Unfortunately I think the dB unit is forever ruined by the custom amongst electronics folks of measuring voltage ratios and stating power ratios (with the assumption of constant impedance), and therefore saying that a dB is 1/20 of a factor of ten. 1/20! I understand why this works, but half the places I see dB described are basically consistent with the bizarre idea that “deci-” means one twentieth.
I would like at this juncture to mention that anybody who posts an explanation of why the 1/20 thing “works” (such as it does) is going to get withering eye rolls at this end of the conversation.
Moving on…
Yet another possibility is using nepers instead of decibels. In my context people have actually heard of nepers and if they are doing their homework they are using them. A downside is that the intent of my graphs is to serve a lookup function, so people actually need to get a numeric result off the paper. Contours are logarithmically spaced to handle a wide dynamic range, not because thinking of the number as a log offers any particular insight. Of course, nepers make it that much harder to wind up with a number, another step in the mental process. Still, for some reason it’s hard to put a finger on, nepers fill me with an enchanted contentment.
I had to go to the store, and was thinking I needed to make just such a post when I came back.
Instead, I’ll just collect my Internet Victory Points and move along… 
OK, I dunno, maybe I still love you.