How do you carry errors for brightness magnitudes I have a value for the relative flux $F_2/F_1$. With some uncertainty value $a$.
If I use the equation to get relative magnitude: $m_1 - m_2 = 2.5 \log_{10}(F_2/F_1)$
How do you now calculate the uncertainty value $a$ for this equation? Do you just do the same equation but applied to the uncertainty value or is there some other alteration you have to make to uncertainty values?
You need to do what is called “propagation of uncertainties”. You can search to get more information on that, but briefly if you have some function $f(x)$ that depends on variable $x$, then the uncertainty $\sigma_x$ on the quantity $x$ is related to the uncertainty on $f$ by
$$
\sigma_f = \sqrt{\left(\frac{\partial f}{\partial x}\right)^2 \sigma_x^2}
$$
Here $f$ is your relative magnitude $m_1 - m_2$, and $x$ is your flux ratio.  So taking that partial derivative will give you the factor that relates the two uncertainties.