How can y be written as a function of x?

[Ad VanderVen]

TL;DR Summary

$x$ is a function $f(\alpha)$ of $\alpha$ en $y$ is a function $g(\alpha)$ of $\alpha$. How can $y$ be written as a function of $x$?

$x$ is a function $f(\alpha)$ of $\alpha$:

$$\displaystyle x\, = \,\ln \left( {{\rm e}^{ 0.6931471806\,{\alpha}^{-1}}}-{{\rm e}^{ 0.2876820724\,{\alpha}^{-1}}} \right)$$

and $y$ is a function $g(\alpha)$ of $\alpha$:
$$\displaystyle y\, = \,\ln \left( {{\rm e}^{ 1.386294361\,{\alpha}^{-1}}}-{{\rm e}^{ 0.6931471806\,{\alpha}^{-1}}}\\
\mbox{} \right)$$

How can $y$ be written as a function of $x$?

 

[anuttarasammyak]

You show parameter representation of x and y.
I am afraid we may not get direct formula of y(x). Or are you satisfied with the abstract formula
$$y=g(f^{-1}(x))$$?

 

[BvU]

If you have the range of $\alpha$ you can fit a polynomial -- or something else, depending on required accuracy ?

$\$

 

[aheight]

Mathematica's $\texttt{NSolve}$ for $\alpha(x)$ returns a set of solutions for $a=1/2$ and $b=3/2$ one of which is
$$
\alpha(x)=\frac{1}{2 \left(\log \left(\frac{\sqrt[3]{\frac{2}{3}}}{\sqrt[3]{\sqrt{3} \sqrt{27 e^{2 x}-4}-9 e^x}}+\frac{\sqrt[3]{\sqrt{3} \sqrt{27 e^{2 x}-4}-9 e^x}}{\sqrt[3]{2} 3^{2/3}}\right)+2 i \pi c_1\right)}\text{ if }c_1\in \mathbb{Z}
$$

then $\displaystyle y(x)=\log\bigg(e^{\frac{c}{\alpha(x)}}-e^{\frac{d}{\alpha(x)}}\bigg)$.

You can rationalize your real coefficients (as NSolve may not be able to solve for alpha with real coefficients) such as:

In[41]:= theA = Rationalize[0.6931471806, 10^-10]
Out[41]= 108926/157147
N[theA,10]
Out[42]= 0.6931471807

and use the rational forms in $\texttt{NSolve}$ and go from there.

 

[Office_Shredder]

Just to point out, this would be a little easier to think about it you didn't obscure the fact that those e , to the crazy decimals are really just 4, 2 and 4/3.

I don't actually have any good ideas for you though.

 

[phinds]

How can $y$ be written as a function of $x$?

If you follow @Office_Shredder's advice and simplify, you get, to at least 3 significant digits (you can do more but I didn't want to bother),

y = .406x