At least in v10.1 [`ContourPlot`](http://reference.wolfram.com/language/ref/ContourPlot.html) doesn't support [`ScalingFunctions`](http://reference.wolfram.com/language/ref/ScalingFunctions.html), but [`ListLinePlot`](http://reference.wolfram.com/language/ref/ListLinePlot.html) does, unofficially.  Therefore this might be of some use.

Using `logscale` from https://mathematica.stackexchange.com/q/78524/121 :

    logify[_][x_ /; x == 0] := 0
    logify[off_][x_] := Sign[x] Max[0, (off + Re@Log@x)/off]
    
    inverse[off_][x_] := Sign[x] Exp[(Abs[x] - 1) off]
    
    logscale[n_] := {logify[n], inverse[n]}
    
    (* additional definition *)
    logscale[n_, m_] := logscale /@ {n, m}

Now:

    cp = ContourPlot[
       729 + x^4 + y^4 + 3 x^2 (-225 + y^2) == 730 y^2, {x, -32, 32}, {y, -34, 34}, 
       MaxRecursion -> 3];
    
    pts = Cases[Normal@cp, Line[x_] :> x, -3];
    
    ListLinePlot[pts, ScalingFunctions -> logscale[2, 2], AspectRatio -> 1]

[![enter image description here][1]][1]

You can change the numeric parameters in `logscale` to get different effects; see the linked post for further examples.


  [1]: https://i.sstatic.net/CGgUr.png