Logarithmic ScalesDate: 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 |
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