Marking the suface on Plot3D

Question

Is there any way to mark a specific line on the Plot3D graph in Log scale. My initial plot is: `

Plot3D[(y/(0.5*y + 2*x)), {x, 1/10^10, 1/10^3}, {y, 0, 4.5/10^6}, 
 ScalingFunctions -> {"Log", "Log", None}, 
 ColorFunctionScaling -> True]

` If is is not in Log scale,I know I can use ParametricPlot3D and use Show to overlay them together. But once I plot it in Log scale, ParametricPlot cannot show in Log scale. What I would like to have is something like this:

    Plot3D[(y/(0.5*y + 2*x)), {x, 1/10^10, 1/10^3}, {y, 0, 4.5/10^6}, 
 ScalingFunctions -> {"Log", "Log", None}, 
 ColorFunctionScaling -> True, 
 MeshFunctions -> Function @@@ {{{v, u}, v^3}}, Mesh -> 1, 
 PlotPoints -> 100, Ticks -> Automatic]

` However, since the MeshFunctions does not give a meaningful idea of what does the line stands for. Is there a way to just specify the X value to plot a line on that surface? Thanks!

Answer 1

The first two coordinates passed to the mesh functions are the Log of the x and y values. Here is the code for my interpretation of the intention of the OP's example MeshFunctions code (with a different value for the mesh line):

Plot3D[(y/(y/2 + 2*x)), {x, 1/10^10, 1/10^3}, {y, 0, 4.5/10^6}, 
 ScalingFunctions -> {"Log", "Log", None}, 
 ColorFunctionScaling -> True, 
 MeshFunctions -> {Function[{logx, logy, z}, Exp[logy]^3]}, 
 Mesh -> {{(5*^-7)^3}}, PlotPoints -> 100, Ticks -> Automatic]