Astronomy’s brightness scale is backwards and logarithmic, both on purpose. Backwards because the Greeks ranked the brightest stars ‘first magnitude’ and the faintest ‘sixth’. Logarithmic because the eye is — so a gap of exactly 5 magnitudes means one star is a full HUNDRED times brighter. Slide the magnitude gap and watch the ratio leap.
The apparent-magnitude relation: m₁ − m₂ = −2.5·log₁₀(F₁/F₂), so brightness ratio = 10^(−Δm/2.5). The scale was DEFINED so that 5 magnitudes equal exactly a factor of 100 in received flux; therefore each single magnitude is the fifth root of 100 ≈ 2.512×. Smaller (even negative) numbers are brighter. A fail-loud self-check throws unless Δm=5 gives a ratio of exactly 100 and Δm=1 gives 2.512.
Apparent magnitude mixes true luminosity with distance (absolute magnitude fixes distance at 10 parsecs); and the eye’s response is only approximately logarithmic (Pogson standardised the scale in 1856). The 10^(−Δm/2.5) arithmetic is exact.