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 reason
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 alot of our perceptual abilities do not
occur in a linear manner. Our perceptions are not tune 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 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. But following that 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 decsribing 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. 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. Again Matlab makes it easy to plot these using teh
functions "semilogy(x,y)" and "loglog(x,y)".