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.