I found :

$$B-V = -2.5 \log \left(\frac{f_B}{f_V}\right)$$

From this question. It works for my purposes but I am not sure I understand where it comes from ? (especially the -2.5 coefficient) Could someone care to elaborate further ?

I found :

$$B-V = -2.5 \log_{10} \left(\frac{f_B}{f_V}\right)$$

From this question. It works for my purposes.

Those are magnitudes on the left. Five magnitudes are a factor of 100, so 2.5 magnitudes are a factor of 10. So if $f_B/f_V =$ 100 (i.e. 10$^2$) then $B-V =$5.