I have to minimize
$ |\mathrm{log}(1-\alpha_1) - \mathrm{log}(\alpha_2)|$
my constraints is $\alpha_1 +\alpha_2 = \alpha$ and $0<\alpha<1$
this is related to finding optimal confidence interval in statistical inference subject my background is statistics so I am very weak in math so please understand my dumb question. but I was wondering if someone could enlighten me with this problem.
edit: actually $ \mathrm{log}(1-\alpha_1) - \mathrm{log}(\alpha_2)$ is the distance so probably I will have minimize the absolute value?