A double star consists of two stars of equal brightness at equal distance from the Earth. What is the apparent magnitude of the combined object? The magnitude of each star in the V-band is mV= 8.34. If these stars are so close together that they appear as one object, what is the apparent magnitude of the combined object?
I just want someone to point me in a general direction of how to approach this problem. The magnitude is logarithmic so it won't be double when combined. Or at least that is what I think. How do I apply the formula
$$m=-2.75 \log_{10} F + C$$
The magnitude formula is $$m = -2.5 \log_{10} \left(\frac {F}{F_{0}}\right)$$
Where $F_0$ is the flux of magnitude 0 star.
So if you have two stars of equal brightness, the flux is doubled, and you get
$$m = -2.5 \log_{10} \left(\frac {2F}{F_{0}}\right)\\
= -2.5\log_{10}(2) -2.5 \log_{10} \left(\frac {F}{F_{0}}\right)$$
The first value is about -0.75, so the double star is 0.75 magnitudes brighter.  If each object is 8.34 then the double star appears as a 7.59 object.