Why Use a Logarithmic Scale to Display Data?
Date: 01/26/2008 at 17:41:06 From: Kristi Subject: logarithmic scales I have read the responses to Brian regarding why someone might use a logarithmic scale: http://mathforum.org/library/drmath/view/55520.html I am 50 and have limited math background but recently read somewhere that log scales help you see data when you are looking at values that range largely. I still can't see it. Can someone give me some steps I can do so I can see what they are trying to say? Also can you help with this statement? I read if the ear did not hear logarithmically that we would only hear very loud sounds. Can you expound on that? I want to try to see it by doing some plotting or something. I just can't understand what is meant. Please include explanation of what I should see in case I still don't make the connection. What a wonderful site! Thank you.
Date: 01/26/2008 at 23:45:05 From: Doctor Peterson Subject: Re: logarithmic scales Hi, Kristi. Let's take an example: the pH of a solution, which indicates the acidity or alkalinity. It is defined as pH = -log[H+] That is, it is the negative of the base-ten logarithm of the concentration of hydrogen ions. Why do we use the logarithm, and not just use the concentration itself? Well, reversing this definition, [H+] = 10^-pH A neutral solution has pH = 7, so its concentration is [H+] = 10^-7 = 0.0000001 A strong acid might have pH = 3, so its concentration is [H+] = 10^-3 = 0.001 A strong base might have pH = 10, so its concentration is [H+] = 10^-10 = 0.0000000001 Now, those are ugly numbers to try to remember, or to recognize when written down. Maybe we could scale them and take our measure of acidity to be 10^7 times the concentration, so that we would have base: 0.001 units neutral: 1 unit acid: 10000 units But still, that's a wide spread--and it could go a lot wider. Suppose we did an experiment on, say, the rate of some reaction in different environments. If our horizontal axis used these units, our three data points would be something like this: +-------------------------------------------------------------+--> 0 10000 ^ ^ | | base acid neutral With the scale set to include the acid, you couldn't distinguish the base from the neutral--or either of those from 0! Yet probably there would be a significant difference in the rate of the reaction. Your graph would be just about unreadable. So you'd try graphing it on a logarithmic scale--very possibly a log-log scale, so that both scales can vary just as widely. Then you could make sense of all the data. Now the acidity scale would look like this: +---+---+---+---+---+---+---+---+---+---+---+---+---+---+---> .001 1 10000 ^ ^ ^ | | | base neutral acid Or, you'd invent the pH, so you had not only a nicer graph, but easier numbers to work with: +---+---+---+---+---+---+---+---+---+---+---+---+---+---+---> 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 ^ ^ ^ | | | acid neutral base The ear works similarly: it can distinguish both loud and soft sounds just the way a logarithmic scale can distinguish large and small numbers, on the same scale. Does that help? - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
Date: 01/27/2008 at 08:38:14 From: Doctor Fenton Subject: Re: logarithmic scales Hi Kristi, Thanks for writing to Dr. Math. There is a pretty good discussion with diagrams at: Wikipedia: Logarithmic scale http://en.wikipedia.org/wiki/Logarithmic_scale Try reading that reference and see if that answers some of your questions. If you have any more questions, please write back and I will try to explain further. - Doctor Fenton, The Math Forum http://mathforum.org/dr.math/
Date: 01/28/2008 at 14:07:25 From: Kristi Subject: Thank you (logarithmic scales) Dear Dr. Peterson, Thank you so much for the wonderful explanation regarding how log scales let us see all the information. It was great. I see it now. Very much obliged. And Dr. Fenton, thanks for the tip on web site. It was helpful, too.
Search the Dr. Math Library:
Ask Dr. MathTM
© 1994- The Math Forum at NCTM. All rights reserved.