How can I solve for $x$ and $y$ in terms of $a$, $b$, and $c$ given these two equations? $a$, $b$, and $c$ are known to be positive real numbers.
$$ b = \exp \Big[-a \cdot (1 + (1+x)^{-y}) \Big] $$ $$ c = \exp \Big[-a \cdot (1 + (1+x)^{-y} + (1 + 2x)^{-y}) \Big] $$
There are two equations and two unknowns so it should in theory be possible, right?
From the given equations, we can isolate the two quantities $$ \begin{aligned} (1+x)^{-y} &=A\ ,\\ (1+2x)^{-y} &=B\ , \end{aligned} $$ where $A,B$ are "easily" written in terms of $a,b,c$. Now we take logarithms and build the quotient, so EDITED $$ \frac {\ln(1+x)}{\ln(1+2x)}=\frac{\ln A}{\ln B}\ . $$ This leads to a transcendental equation in $x$. This determines $x$. From the first equation above, say, we obtain $y$.
Note: Many thanks go to g-kov for pointing out an awful error...