Magnitude scale theory

How bright a star looks is given by its apparent magnitude. The absolute magnitude of a star is defined as the apparent magnitude that it would have if placed at a distance of 10 parsecs from the Earth.

Consider two stars where star A appears to be brighter than star B.
Let the apparent magnitude of star A = m_A and the apparent magnitude of star B be m_B.
IA/IB = 100(m_B – m_A)/5

Therefore taking logs of both sides : m_B – m_A = 5/2[lg(I_A/I_B)]

Now let the magnitude of A (m_A) be that at 10 parsecs, in other words the absolute magnitude of the star (M) and let m_B be the magnitude (m) at some other distance d (also measured in parsecs).

Therefore: lots m – M = 5/2[lg(I_A/I_B)] lots of space

M – m = 5 - 5lg(d)