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The Greek astronomer Hipparchus is usually credited with the origin of the magnitude scale. He assigned the brightest stars he could see with his eye a magnitude of $1$ and the faintest a magnitude of $6$. However, in terms of the amount of energy received, a sixth magnitude star is not $6\times$ fainter than a first magnitude star, but more like $100\times$ fainter, due to the non-linear response of the human eye to light.

This led the English astronomer Norman Pogson to formalize the magnitude system in 1856. He proposed that a sixth magnitude star should be precisely $100\times$ fainter than a first magnitude star, so that each magnitude corresponds to a change in brightness of $100^{1/(6-1)} = 2.512$. For example, a star of magnitude $2$ is $2.512^1\times=2.512\times$ times fainter than a star of magnitude $1$, a star of magnitude $6$ is $2.512^2\times=6.3\times$ fainter than a star of magnitude $4$, and a star of magnitude $25$ is $2.512^5\times=100\times$ fainter than a star of magnitude $20$. ¹⁾

Hence, Pogson's ratio of $2.512$ leads us to Pogson's equation:

$$\frac{F_1}{F_2} = \biggl(100^{1/5} \biggr)^{-(m_1-m_2)}$$

where $F_1$ and $F_2$ are the fluxes of two stars, $m_1$ and $m_2$ are their magnitudes, and the minus sign in front of the exponent accounts for the fact that numerically larger magnitudes refer to fainter stars²⁾.

Taking logarithms of Pogson's equation, we obtain:

$$\log_{10}\frac{F_1}{F_2} = -(m_1-m_2) \cdot \log_{10}(100^{1/5}) = -\frac{2}{5} (m_1-m_2)$$

More conveniently, we can write:

$$\frac{F_1}{F_2} = 10^{-\frac{2}{5}(m_1-m_2)}$$ and

$$m_1-m_2 = -2.5 \log_{10}\frac{F_1}{F_2}$$

Sources

1. Lifted with minor modifications from: http://www.vikdhillon.staff.shef.ac.uk/teaching/phy217/instruments/phy217_inst_mags.html