How to compare variables with different units?

The easiest way is to mean-center each set of data. You do that by calculating the mean (average) of the original data in that set, then you divide each data point in that set by that mean. This scales the data around 1 (the mean divided by the mean equals one) so that data with very different units can be shown on the same graph. Usually one does this with only the y-values, but it can be done for x-values if the spread of data is large (e.g, comparing data on minutes scale with data on years scale).

The advantage of mean-centering over log or log-log transforms is that the original variability in the data is retained. Taking the log compresses the range of the data and artificially smooths the variability out. (A log-log plot of data that is actually a parabola will give a straight line with r^2>0.99.)

Mean-centering is used in chemometrics, so that's how I know about it (I'm a chemist).

EDIT

"Additional Details

I am only doing basic stats. What I really need is to know what label to put on the graph. Thanks"

O.K., sorry for the overkill!

What I have seen in the past is this: on the left side of the graph, the vertical axis is drawn with the first set of units indicated, and on the right side of the graph, another vertical axis is drawn with the other set of units indicated.

Here's an example of that:

http://www.rfcafe.com/references/electrical/ew-rad...

Just write log(mm/yr) for rainfall, you should note the base you have used for the logarithms (natural logs or log base 10)