Measurements on a Logarithmic Scale

Date: 12/13/98 at 15:27:07
From: Tori
Subject: logarithms

I have been asked to find how a logarithmic scale is used to measure 
some quantity such as star brightness, earthquake intensity, or 
acidity. I also have to include information on what physical 
characteristics underlie the quantity, how the quantity is measured,  
how it is "transformed" to a logarithmic scale, and why or how the 
logarithmic scale is more useful than the original scale.  

If you could help me, I'd really appreciate it. Thanks so much!

Date: 12/14/98 at 12:06:18
From: Doctor Peterson
Subject: Re: logarithms

Hi, Tori. I just did a quick search for pages including both 
"logarithm" and either "pH" or "richter". Here are couple of places I 
found; you can find more:

   http://bcn.boulder.co.us/basin/data/COBWQ/info/pH.html   
   (on pH -- from Sheila Murphy, Research Analyst, BASIN project, 
    City of Boulder)

   http://www.seismo.unr.edu/ftp/pub/louie/class/100/magnitude.html   
   (on the Richter scale -- from John N. Louie)

You might also want to investigate the decibel scale for loudness. 
Here's a page and a quote from it as to why a logarithmic scale is 
useful:

   http://www.point-and-click.com/Campanella_Acoustics/faq/faq.htm   
   (from Campanella Associates) [Note page is now gone; 
    archived version available]

   1) Quantities of interest often exhibit such huge ranges of
      variation that a dB scale is more convenient than a linear 
      scale. For example, sound pressure radiated by a submarine may 
      vary by eight orders of magnitude depending on direction.

   2) The human ear interprets loudness more easily interpreted with
      a logarithmic scale than with a linear scale.

I can't answer all your questions about the science, but in general, 
as this quote indicates, we use a logarithmic scale when there is a 
wide range of values, and when the significance of a change in that 
value depends not on the absolute size of the change but on the size 
of the change in proportion to the value itself. If adding 1 to a 
value is just as big a change whether the original value was 1 or 
1000, a linear scale makes sense. If doubling a value is just as big a 
change whether it is from 1 to 2 or from 1000 to 2000, a logarithmic 
scale is appropriate.

For instance, pH is the negative of the logarithm of the concentration 
of hydrogen (or hydronium) ions. If we just used the concentration as 
our measure of acidity, then an acid with pH 3 would be said to have a 
concentration of .001 (10^-3), and a weaker acid with pH 4 would be 
0.0001. Neutral water would be 0.0000001, and a strong base with pH 11 
would be 0.00000000001. Try graphing these (that is, write them along 
the x axis), and you'll find that it's impossible to show all these 
values on the same graph! If you can see acids at all, neutral and 
basic values will be indistinguishable. You might be able to put 0.001 
and 0.0001 on your axis, but 0.0000001 will look like zero, and 0.01 
(a really strong acid) will be off your paper. If the difference 
between 0.000001 and 0.00001 were no more significant than between 
0.001 and 0.001001, then this would be appropriate. The tiny values 
wouldn't be worth showing. But because it's more important to 
distinguish the former two (weak and weaker acids) than the latter two 
(practically the same strong acid), we need a way to show all these 
values on our graph. A logarithmic scale allows us to do this.

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