Plotting logarithmic scale graphs

Not entirely. Yes, you have to describe the relation between rho and E, but.

You have to realise that for every value of rho, there are 0, 1 or 2 values of E that describe the outer perimeter of each of the shapes you want to draw on the graph.

Let's take a simple example: how do you draw a circle on the x, y plane. It can be done like this:

This describes circle with radius 1, and the centre is at x=2, y=2.

For each value of x, two values are calculated.

Now we can run x from 1 to 3 and plot the two components of Circle().

If we now change the axes of the plot to log we get:

If you want to convey the impression of a single curve, you can change the blue trace to red.

There is an alternative way, using another parameter, but I don't know if you (can) have that in your example. Anyway, for a circle it can go like this:

And with log scales:

Success!
Luc