Calculating $\frac{e^{\frac{z_1}{T}}}{e^{\frac{z_1}{T}} + e^{\frac{z_2}{T}}}$ from $\frac{e^{z_1}}{e^{z_1} + e^{z_2}}$

Question

Let's say I know the value of

$\frac{e^{z_1}}{e^{z_1} + e^{z_2}} = x $ (say)

Now I want to get the value of

$\frac{e^{\frac{z_1}{T}}}{e^{\frac{z_1}{T}} + e^{\frac{z_2}{T}}} = y $ (say), where $T \epsilon \mathbb{Z^+}$, some integer constant.

Now, is there some way to get the value of $y$ directly from $x$, without knowing or calculating the intermediate values of $e^{z_1}$ or $e^{\frac{z_1}{T}}$?

Answer 1

Yes you can. Rewrite $x$ as $\dfrac1{1+e^{z_2-z_1}}=\dfrac1{1+w}$.

Then similarly $$y=\dfrac1{1+\sqrt[T]w}=\frac1{1+\sqrt[T]{\frac1x-1}}.$$