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## Main Question or Discussion Point

Say we have a decay of the form

[tex]A e^{-a x} + B e^{-b x}[/tex].

I haven't had much luck trying to calculate the half-life of such a decay (I'm not sure it's possible, analytically), i.e. solve

[tex]A e^{-a x} + B e^{-b x} = \frac{A+B}{2}[/tex].

However, if that's not possible, I'm wondering whether there might still be a way to prove that if the above decay has half-life [tex]t[/tex], then a decay given by

[tex]A e^{-a c x} + B e^{-b c x}[/tex]

has half-life [tex]\frac{t}{c}[/tex]. This seems to be true empirically and would make sense, I think. Does anyone have an idea how one might prove it to be true, though?

[tex]A e^{-a x} + B e^{-b x}[/tex].

I haven't had much luck trying to calculate the half-life of such a decay (I'm not sure it's possible, analytically), i.e. solve

[tex]A e^{-a x} + B e^{-b x} = \frac{A+B}{2}[/tex].

However, if that's not possible, I'm wondering whether there might still be a way to prove that if the above decay has half-life [tex]t[/tex], then a decay given by

[tex]A e^{-a c x} + B e^{-b c x}[/tex]

has half-life [tex]\frac{t}{c}[/tex]. This seems to be true empirically and would make sense, I think. Does anyone have an idea how one might prove it to be true, though?