11 Answers. Yes, logarithms always give dimensionless numbers, but no, it's not physical to take the logarithm of anything with units.

Do logs have units?

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.

What happens to units in log?

The units of a ln(p) would generally be referred to as "log Pa" or "log atm." Taking the logarithm doesn't actually change the dimension of the argument at all -- the logarithm of pressure is still pressure -- but it does change the numerical value, and thus "Pa" and "log Pa" should be considered different units.