Level curve with function containing natural logarithm

Question

How do I sketch a level curve that has natural logarithm in its function? For example:

$Z(x, y) = ln(xy) − x$ when $x > 0$ and $y > 0$

I can't find anything about it, so if you have a source I could read or try to explain it here it would be great.

Answer 1

Note that\begin{align}\log(xy)-x=C&\iff\log(xy)=x+C\\&\iff xy=e^xe^C\\&\iff y=e^C\frac{e^x}x.\end{align}So, your level curve is just the graph of the map $x\mapsto e^C\dfrac{e^x}x$.