Exponentials, logarithms and rescaling of data In many cases if you were to plot data points obtained from biological measurements (for example mean brain weight as compared to mean total body weight for mammals of various sizes) you would find that the data do not at all fall on a straight line. There are a variety of reasons for this. One example is illustrated by the way we hear sounds. Imagine yourself in a classroom with one other person in it and they drop a pencil (assume the floor doesn't have a rug). You would likely hear the pencil drop. Now imagine the same person dropping the pencil with the classroom full of people talking before class starts. You likely would not hear the pencil fall at all, but it certainly is making the same "sound" (e.g. the physics of the situation hasn't changed). Why does this happen? It is because our hearing is better at detecting "relative" rather than absolute differences between sound levels. Of course this is very simplified since it also depends upon the frequency of the sound, but in general much of our perceptual abilities do not occur in a linear manner. Our perceptions are not tuned to detect "additive differences" but rather to detect "multiplicative differences". Another example is given by population growth. Imagine algae growing in a petri dish, starting from a single cell. Through time the cell present will split, then each cell will split again then split again, so that the total population of cells doesn't increase linearly (in an additive manner) through time but multiplicatively (by doubling). If you were to plot the number of cells through time, it would increase geometrically, not linearly. The above are two examples why exponentials and logarithms are used so much in biology. Exponentials are used to describe something which increases (or decreases) in a multiplicative manner. Logarithms are a way to rescale something which is increasing (or decreasing) in a multiplicative manner so as to make it increase (or decrease) linearly. This arises, as you will recall, from the fact that logarithms turn multiplication into addition - the log of a product is the sum of the logs of the components of the product - log(a*b) = log(a) + lob(b). This would be a good time to refresh your memory of logs and exponentials by reading Sections 5 and 15 of the text. Allometry An extremely common relationship that arises over and over again in biology is the notion of an allometric relationship between two measurements. x and y are said to be allometrically related if y = a*x^b where a and b are constants. For a good explanation of allometry work through the module BODY SIZE CONSTRAINTS IN XYLEM-SUCKING INSECTS: ALLOMETRIC RELATIONSHIPS posted from the course home page under the Modules section available at http://www.tiem.utk.edu/~gross/bioed/bealsmodules/allometry.html and the module on SPECIES-AREA RELATIONSHIPS posted at http://www.tiem.utk.edu/~gross/bioed/bealsmodules/spec_area.html Log-log and Semilog graphs How do we tell if a non-linear relationship is a better model for how two datasets are related than a linear one? Of course first you should do a scatter plot of the data. If there appears to be a pattern in the data, so that for example one variable tends to increase (or decrease) as the other increases, but a line does not seem to be a good "fit" to the data, then the next step is to try to "transform" the data. In this case, you should try plotting the data on a semi-log or a log-log plot if you have any reason to suspect that the data are related allometricly (use a log-log plot) or if one appears to be geometrically changing with the other (e.g. there is an exponential function describing them such as y = a * b^x where a and b are constants). By taking logarithms on both sides of an allometric relationship, you will see that the logarithms of the measurements are linearly related. Suppose that y = a * x^b Then taking logarithms of both sides gives log (y) = log (a) + b * log (x) which says that log (y) plotted versus log (x) is a straight line with slope b and vertical intercept (when log(x) = 0 which occurs when x=1) at log (a). This means that when you plot y vs. x using a log-log plot, you get a straight line. Similarly, by taking logarithms on both sides of a geometric relationship, you will see that the logarithm of one of the measurements is linearly related to the other measurement. Suppose that y = a * b^x Then taking logarithms of both sides gives log (y) = log (a) + x * log(b) which says that log(y) plotted versus x is straight line with slope log(b) and vertical intercept (when x = 0) at log (a). This means that when you plot y vs. x using a semilog plot, you get a straight line. Matlab makes it easy to plot these using the functions "semilogy(x,y)" and "loglog(x,y)".