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)".