So I have never actually found a good answer or even a good resource which discusses this so I appeal to experts here at stack exchange because this problem came up again today. What happens to the units of a physical quantity after I take its (natural) logarithm. Suppose I am working with some measured data and the units are Volts. Then I want to plot the time series on a log-scale, only the ordinate is on the log scale, not the abscissa. So the x-axis is definitely in time (seconds let's say) but what are the units on the y-axis? Will it be Volts or log(Volts) or something? If I square the quantities, then the units are squared too so what if I take the log? A rationale in addition to the answer will be appreciated as well.
I guess whatever the answer, the same goes for taking the exponential or sine of the data too, right?
Best Answer
Overall, the argument $x$ of $\ln(x)$ must be unitless, and a log transformed quantity must be unitless. If $x = 0.5$ is measured in some units, say, seconds, then taking the log actually means $\ln(0.5s/1s) = \ln(0.5)$. See this for more information about other transcendental functions. Hope this helps.