\[m_2-m_1=-2.5 log_{10} (\frac{b_2}{b_1})\]
.In this equation
\[m_1, \: m_2\]
are the apparent magnitudes of two stars and \[b_1, \: b_2\]
their apparent brightnesses (if the power output of a star at a distance \[d\]
is \[L\]
then \[b= \frac{L}{4 \pi d^2}\]
).Suppose then that the apparent brightnesses of two stars differs by a magnitude of 1, so that
\[m_2-m_1=1\]
, then\[1=-2.5 log_{10} (\frac{b_2}{b_1})\]
\[log_{10} (\frac{b_2}{b_1})= \frac{1}{-2.5}=- \frac{2}{5}\]
\[\frac{b_2}{b_1}=10^{- \frac{2}{5}} \simeq \frac{1}{2.512}\]
Hence
\[b_1=2.512b_2\]
.Note that
\[m_2 \gt m_1 \rightarrow b_1 \gt b_2\]
.