Re: [Phys-l] use of lgarithms

[John Mallinckrodt <ajm@csupomona.edu>]

Here are two possible ways of thinking about it.

1. First of all it is important to note that one simply cannot take  
the log of something like "10 meters."  If you think you can, then  
tell me if the result is the same as the log of "1000 cm."  So you  
are already implicitly dealing with the units in the following manner:

Rewrite the equation x = (1/2)gt^2 as

    x/ul = (g/2)/(ul/ut^2) * (t/ut)^2

where ul and ut are arbitrary units of length and time.  For  
instance, you might choose ul = 1 meter and ut = 1 sec, but any other  
choice would also work.  Now you can take the logs.  What you get is  
really nothing more than what you are actually already doing,

  log(x/ul) = 2 log(t/ut) + log[(g/2)/(ul/ut^2)]

2. Alternatively, consider the act of finding the slope by hand.  You  
pick two points on the best fit straight line and find

  slope =  [log(x2) - log(x1)] / [log(t2) - log(t1)] = log (x2/x1) /  
log(t2/t1)

In other words, the process of finding the slope involves taking the  
logs of unitless *ratios* of lengths and times so there is no problem.

John Mallinckrodt
Cal Poly Pomona

On Oct 28, 2009, at 8:09 AM, LaMontagne, Bob wrote:

> Hi All,
>
> I have been on this list for a long time - too long for many of you  
> - however I don't recall the following issue being dealt with.
>
> We have a lab where students release a ball from rest and let it  
> fall on a contact sensitive timing switch. They can produce a table  
> of distance fallen and time to fall.
>
> For many years, we have had them analyze the data by making a plot  
> on log-log paper of x versus t. Since x=0.5gt^2, the plot will have  
> a slope of 2. The students verify this. They also find the value of  
> g (actually g/2) by finding the intercept with the vertical line  
> where log(t)=0, i.e., where t=1. The value of g found this way is  
> usually reasonable.
>
> My question to the group is about units. The essence of the plot is
>
> Log(x) = 2 log(t) + log(g/2)
>
> How do we reconcile units? The slope of 2 is simple because it is  
> dimensionless. But what about log(x)? The argument of a logarithm  
> appears to have to be dimensionless (The log of x contains x^2,  
> x^3, etc, so dimensioning x produces every power of the dimension).
>
> We have the students get the numerical value of g from the  
> intercept - but what about the units?
>
> For RC circuits this is not an issue because we can form log(i/i0)=- 
> t/RC and the argument of the log is dimensionless.
>
> Bob at PC
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