Why is the base a = 2.512 in Pogson’s formula for connecting illumination from space objects with their stellar values?

In the magnitude scale introduced by Hipparchus, the base a turned out to be 2.5. The modern value of a is chosen to be 2.512, since the logarithm of this value is 0.4. This leads to a simpler form of Pogson’s formula.
The magnitude scale is determined so that a 5-magnitude change in brightness corresponds to a 100-fold change in illumination. Therefore, the coefficient in Pogson’s formula is 5 / lg100 = 5/2 = 2.5.

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