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I am trying to find the form of $z=f(x,y)$. I know that:

Contour lines in the $XZ$ plane are of the form: $z=A*ln(x)+B$
(the $A$ and $B$ parameters vary with $y$)

Contour lines in the $YZ$ plane are of the form: $z=Cy^2+Dy+E$
(the $C$, $D$ and $E$ parameters vary with $x$)

What is then the form of $z=f(x,y)$ ?

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I'm not sure what you mean by $A$ and $B$ varying with $y$ in the context of contour lines in the $XZ$ plane (and likewise for $x$ and $YZ$). I'll assume what you mean is that there is a function $z=f(x,y)$ and the first equation describes the relationship between $x$ and $z$ if you hold $y$ fixed and the second equation describes the relationship between $y$ and $z$ if you hold $x$ fixed.

Since the two expressions for $z$ have to be equal, the dependency of the parameters on $y$ in the first equation must be such that a quadratic function of $y$ results for all $x$. Since $\ln x$ takes different values for different $x$, this can only happen if $A$ and $B$ are both quadratic functions of $y$. Thus, the most general form for $z$ compatible with the given conditions is

$z=(a_2y^2+a_1y+a_0)\ln x+b_2y^2+b_1y+b_0$

with

$ \begin{eqnarray} A&=&a_2y^2+a_1y+a_0\;,\\ B&=&b_2y^2+b_1y+b_0\;,\\ C&=&a_2\ln x+b_2\;,\\ D&=&a_1\ln x+b_1\;,\\ E&=&a_0\ln x+b_0\;.\\ \end{eqnarray} $