I keep forgetting this sort of simple algebra (aging is no fun), so here’s a note.
Define magnitude 
to be related to flux 
as follows:
where 
is the zero point flux which defines the magnitude scale. Let 
be the magnitude uncertainty, which is related to the fractional uncertainty in flux, 
. We wish to find how 
is related to 
. This means
![\begin{array}{rcl} \pm \Delta m &=& -2.5 \log{\left[\left(1 \mp \frac{\Delta f}{f}\right)\frac{f}{f_0}\right]} + 2.5 \log{\frac{f}{f_0}} \\ &=& -2.5 \log{\left(1 \mp \frac{\Delta f}{f}\right)} \end{array} \ .](http://s0.wp.com/latex.php?latex=%5Cbegin%7Barray%7D%7Brcl%7D+%5Cpm+%5CDelta+m+%26%3D%26+-2.5+%5Clog%7B%5Cleft%5B%5Cleft%281+%5Cmp+%5Cfrac%7B%5CDelta+f%7D%7Bf%7D%5Cright%29%5Cfrac%7Bf%7D%7Bf_0%7D%5Cright%5D%7D+%2B+2.5+%5Clog%7B%5Cfrac%7Bf%7D%7Bf_0%7D%7D+%5C%5C+%26%3D%26+-2.5+%5Clog%7B%5Cleft%281+%5Cmp+%5Cfrac%7B%5CDelta+f%7D%7Bf%7D%5Cright%29%7D+%5Cend%7Barray%7D+%5C+.+&bg=ffffff&fg=000000&s=0 "\begin{array}{rcl} \pm \Delta m &=& -2.5 \log{\left[\left(1 \mp \frac{\Delta f}{f}\right)\frac{f}{f_0}\right]} + 2.5 \log{\frac{f}{f_0}} \\ &=& -2.5 \log{\left(1 \mp \frac{\Delta f}{f}\right)} \end{array} \ .")
Solving for we get
In summary, we have the following relationships:
When 
or 
, we may expand the log function about 1 and retain only the first order term:
for 
. In this limit we can write 
. Hence
This is why an uncertainty in magnitude, when small, is often an adequate approximation to the fractional uncertainty in flux.
