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.