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

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

##\ ##
 
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:

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

and use the rational forms in ##\texttt{NSolve}## and go from there.
 
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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.
 
Ad VanderVen said:
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
 
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