Logarithms and units

One of the things that every intro calculus student learns is: $${d\ln x\over dx} = {1\over x}$$ This property of the logarithm leads to something else, which turns out to be useful to physicists and astronomers, but is never explicitly taught. If we rearrange this equation to read $${d\ln x} = {dx\over x}$$ we see that a given change in the logarithm ($d\ln x$) corresponds to a given fractional change in x. This equation also implies that the logarithm of anything is unitless, as follows:

• the right side of this equation, ${dx\over x}$, is unitless regardless of the units of x;
• therefore the left side, $d\ln x$, must also be unitless; 
• $d\ln x$ must have the same units as $\ln x$;
• therefore $\ln x$ must also be unitless, regardless of the units of x. 

Physics students keeping track of their units can be stumped: what units does the log of a current or a voltage have? This tiny bit of math helps us see that the answer is "none."

The fact that $d \ln x$ specifies a fractional change in x has further repercussions in astronomy, because it is traditional to quote the measurement of a flux $f$ in the magnitude system: $$m = -2.5 \log_{10} {f\over f_0}$$ where $f_0$ is some reference flux. This means that a quoted uncertainty in the magnitude of a star or galaxy, $dm$, specifies a fractional uncertainty in the flux. Let's work out the details: $\log_{10} x$ is the same as ${\ln x \over \ln 10}$ so $$dm = -{2.5\over \ln 10}  d\ln{f\over f_0}$$ $$dm = -{2.5\over \ln 10} {df\over f}$$ Because $\ln 10\approx 2.30$, we get $dm \approx -1.086 {df\over f}$.  For quick estimation purposes, the magnitude uncertainty is about the same as the fractional uncertainty in flux.

This explains why a 0.1 mag uncertainty is about a 10% flux uncertainty, regardless of the magnitude. One should not say that a 0.1 mag uncertainty is a 1% uncertainty in an $m=10$ star, nor a 0.5% uncertainty in an $m=20$ galaxy.  For the quantity that matters---the flux of the object---a 0.1 mag uncertainty implies about a 10% uncertainty regardless of the flux.