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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 [1]