A math terminology question

October 27, 2006 12:27 PM

A math terminology question. What is meant by a "log-linear" relationship, and what distinguishes it from a plain logarithmic one?
posted by shoos to writing & language (13 comments total)

Could it be a linear graph where the x-axis is "x" and the y-axis is "log(y)"? If this is the case, then there isn't any real difference; just a difference in how you perceive it.
posted by sbutler at 12:30 PM on October 27, 2006

What is the context? Is it possibly presented as a choice (either/or) e.g., at the top of a stock chart?
posted by weapons-grade pandemonium at 12:42 PM on October 27, 2006

The context is scientific journal articles. There are claims that the relationship between two variables (say a drug dose and a response) is "log-linear."
posted by shoos at 12:52 PM on October 27, 2006

Seconding sbutler's suggestion. Log-Linear refers to a graph that's logarithmic on the y-axis and linear on the x-axis. (Usually this is used to distinguish from a Log-Log plot, which would be logarithmic on both axes.)

on preview: Based on your comment, it means the relationship would look like a line on such a set of axes.
posted by dseaton at 12:53 PM on October 27, 2006

this is usually how I've heard them differentiated, but I was also in econ.
posted by milkrate at 12:53 PM on October 27, 2006

In that context then it means that once converted into logarithms their is a linear relationship between dose and response, or that the efficacy of dosage grows geometrically not arithmetically.
posted by JPD at 1:02 PM on October 27, 2006

Thanks for the suggestions.

One thing that has gotten me confused is that the graph below was described by its authors as showing a log-linear relationship (or two, rather). But they both appear to be linear relationships to me! (despite the log scales). That can't be right, can it?

graph
posted by shoos at 1:55 PM on October 27, 2006

We can't view that image.
posted by ikkyu2 at 2:31 PM on October 27, 2006

Ah, blogger's disabled metafilter as a referer. It can be viewed by copying and manually pasting the link.

When you say that something has a relationship, you can say "y goes as x." or "y goes as three times x." or "y goes as the square of x" or even "y goes as the cube root of x."

Implicit in this is that if you plot y on the y axis and "what y goes as" on the x axis, you see a line (more or less.)

In this case the scale of x and y are both logarithmic. So instead of saying "y goes as x," you're getting the mathematical equivalent: "log (y) goes as log (x)." In other words, instead of plotting on a graph with 0, 1, 2, 3 as the axis, we are seeing 10^0, 10^1, 10^2, 10^3. And we're seeing it on each axis.

You couldn't easily see this kind of relationship on a standard linear plot, because all but one or two points would be crammed down into the lower left corner where they can't be distinguished from the origin. So this is a way of displaying a linear relationship that lets you see all the different data points in relation to thet others. That's why they're calling it log-linear: the scale is logarithmic and the relationship is linear.
posted by ikkyu2 at 2:38 PM on October 27, 2006

One point I'd like to make:

In his Nobel Prize biography, Herb Kroemer quotes Fritz Houtermans as having said, "On a double-log plot, my grandmother fits on a straight line."

While ikkyu2 is completely correct in his assessment, it's worth noting that an approximately-linear relationship on a log-log plot is fairly easy to achieve and should not be taken too terribly seriously.
posted by JMOZ at 8:49 PM on October 27, 2006

just to expand a bit on ikkyu2's answer: if you have some function (or dataset) plotted on a log-log plot (that is, one where both axes are logarithmic) and the slope looks to be a line, then the actual relationship between the 2 variables is a power law.

if you superimpose some arbitrary linear scale over the axes of your graph and use that to measure the slope, the slope will tell you the exponent in the power law.
posted by sergeant sandwich at 6:00 AM on October 28, 2006

the slope looks to be a line

whoops, i meant the curve.
posted by sergeant sandwich at 6:03 AM on October 28, 2006

sorry, what i wrote just now isn't quite correct; the linear scale you want to superimpose isn't arbitrary.

what you want to do is: at every tick mark on the plot, replace the number there by its exponent. so 10 becomes 1, 100 becomes 2, 1000 becomes 3, and so on. then you measure the slope in terms of that scale and that will show you the power relationship of the 2 variables.

for the example you showed, the slope of the top curve looks like it's about 1.33 = 4/3, so i would say the relationship goes something like y = x^4/3.
posted by sergeant sandwich at 6:16 AM on October 28, 2006