When dealing with reaction rates for first order reactions, my textbook has this formula:

$\displaystyle \ln [A_f] = -kt + \ln [A_0] $

where $\displaystyle [A_f],[A_0]$ are the final and initial concentrations or pressures of a reactant, and k is a constant with units of 1/sec. This is derived from

$\displaystyle \frac{[A_f]}{[A_0]} = e^{-kt}$

Is there any meaning in asking what the logarithm of a unit of concentration or pressure is? In this case it looks like the inputs to ln are actually dimensionless since the units cancel in the second equation above. In general though, what does a logarithm or exponential function output when given a unit, like $\displaystyle \ln(20 mi/hr)$, $\displaystyle e^{12 Joules}$, or $\displaystyle \ln(5 ArbitraryUnits)$? Do we have to leave it as something like a real number plus ln(mi) - ln(hr) ?