Why $xyz = e^x$ can be seen as the level surface $f(x,y,z) = xyz - e^x$?

Question

That does not make sense to me. I recognize a level surface from the form $f(x,y,z) = k$. Where is the $k$ there? It looks just like a $3$ variables function to me.

Answer 1

Let we have a well-behaved function, say $f(x,y,z)$, so the equation $f(x,y,z)=c$ in whch $c$ is an arbitrary real value defines a surface for each value for $c$. Now, if you assign all possible values $c$ such that the latter equation have any solution, so you'll indeed have a family of surfaces, called level surfaces of that $f$. This is exactly what @sranthrop noted. In fact, if we set $c=0$ when we set $f=c$ then we get $xyz=e^x$. The following plots are parts of whole shape of the functions for values $c=1,2,3,4,5$.

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