Logarithmic Scales

Date: 01/16/2002 at 15:41:17
From: Brian Gove
Subject: Logarithmic scales

In my profession (environmental science) I often see data presented on 
a logarithmic scale. For instance, a particular toxicant concentration 
(i.e., independent variable) plotted against against a specific 
biological response to an organism (i.e., dependent variable) is often 
plotted on a log-log scale. Please explain to me why empirical 
scientific data might be transformed to logarithmic data and 
presented on a logarithmic scale.

I have asked scientists in my agency to explain this to me and have 
never quite understood their answers. I have reviewed several archived 
questions from the Dr. Math site regarding logarithms, but still do 
not really understand the practice of plotting data on a log-log 
scale. 

Thanks for your help!

Date: 01/16/2002 at 17:09:01
From: Doctor Peterson
Subject: Re: logarithmic scales

Hi, Brian.

Looking in our archives, I found this one example of the use of log-
log, which may be interesting to you:

   Finding a Formula That Fits the Data
   http://mathforum.org/dr.math/problems/mullen.1.26.00.html   

On the Web, I located this PDF file of a nice paper on the topic:

   Equations of Straight Lines on Various Graph Papers 
    This document describes step-by-step how to determine the 
    equations of straight lines plotted on linear, semi-log, and 
    log-log graph paper. Includes diagrams and examples.

   http://www.humboldt.edu/~geodept/geology531/531_handouts/equations_of_graphs.pdf   

I'll give a quick explanation.

First, let's look at a semi-log graph. Here, the horizontal axis 
represents x as usual, but the vertical position is not y units from 
the axis but log(y), which I'll call Y to make notation easier. (You 
can use any base you want for the log, but I'll assume base ten.) If 
you draw a straight line on this graph, then it has an equation of the 
form

    Y = ax + b

which means

    log(y) = ax + b

Now, if you raise 10 to the power on each side of this equation, you 
get

    y = 10^(ax + b)

      = 10^(ax) * 10^b

      = k 10^(ax)

where k = 10^b. So if you expect two variables to have an exponential 
relationship, you just have to plot them on semi-log paper, find the 
best-fit line, and use its slope and intercept to find the parameters 
for the equation.

The same sort of thing happens with a log-log graph. Here the 
horizontal position is X = log(x) and the vertical position is 
Y = log(y), so a straight line represents

    Y = aX + b

    log(y) = a log(x) + b

Raising 10 to each power again, we get

    y = 10^(a log(x) * 10^b

      = 10^b (10^log(x))^a

      = k x^a

where k = 10^b. So if x and y are related by a power law or this form, 
you can find the parameters by looking at the slope and intercept.

Look at the situations in which each kind of graph is used, and 
compare it to the equations involved, and you should see exponential 
and power laws.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/   

Date: 01/16/2002 at 17:13:21
From: Doctor Schwa
Subject: Re: Logarithmic scales

Plotting on a log-log scale has many advantages for environmental
science.

1) You are often dealing with very big or very small numbers, and
particularly you are often dealing with numbers that range over many 
orders of magnitude.

For instance, if you are measuring concentrations of some pollutant, 
and your data are something like 0.01 ppb, 0.2 ppb, 2ppb, 117 ppb,
if you plot it on a linear scale, then the three smaller data points
will all be "squished" in at almost the origin while the one big point 
sticks way out. You won't be able to see your data at all.

2) Power laws are very common.

If you plot any power law (y = a * x^b) on a log-log scale, you get a 
straight line whose slope is b. So it's always good to plot power law 
relationships on log-log axes, so you can see how closely they fit to 
a straight line.

On standard axes, power law and exponential relations often look very 
similar. On log-log axes, the power law looks very much like a 
straight line while the exponential relation does not (to make the 
exponential linear, you'll have to take the log of only the y axis).

If the biological response, in your example, is related to some power 
of the toxicant concentration, you'll get a nice straight line. If 
it's another function (logistic, or exponential) those shapes are also 
often easier to recognize on log-log scales (but not so easy to 
describe in words).

3) Often, percent change is what is important.

On standard axes, an equal QUANTITY of change is an equal amount of 
space. On log axes, an equal PERCENT change is an equal amount of 
space. You can thus use your eyes to quickly compare percent growth or 
percent change, instead of having to correct for the magnitude of the
data.

That is, 10 and 12 will be equally far apart as 50 and 60 on log axes, 
because each is a 20% increase.

I hope those ideas help!

- Doctor Schwa, The Math Forum
  http://mathforum.org/dr.math/   

Date: 01/17/2002 at 10:33:12
From: Brian Gove
Subject: Logarithmic scales

Dr. Schwa:
Thank you very much for your excellent response to my question. I see 
now that there are several good reasons why so much of the scientific 
data I see is presented on a log scale. Your explanation was very 
helpful to me and is much appreciated! Thanks again.

Dr. Peterson:
Thank you for your response and the additional sources you have 
provided to help me understand the reasons for presenting scientific 
data on logarithmic graphs. This information has been very helpful 
and I do appreciate your time and knowledge. Thanks again!

Brian Gove