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Using Mathematica 11.0.1.0 on a MacBook Pro (OSX 10.11.6) I tried this:

Clear[x, y, reg1]
reg1 = ImplicitRegion[
   Log[10, 1 + x^2 + y^2] <= 1 + Log[10, x + y], {x, y}];
RegionPlot[reg1]

But got only this output:

RegionPlot[reg1]

But I did get the area.

Area[reg1]

Which gave the correct answer $49\pi$. Then I tried:

RegionPlot[
 Log[10, 1 + x^2 + y^2] <= 1 + Log[10, x + y], {x, -2, 12}, {y, -2, 
  12}]

And I did get the correct image:

enter image description here

Is the fact that RegionPlot[reg1] did not work a bug that should be reported, or have I made some sort of mistake?

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  • 4
    $\begingroup$ RegionPlot[region] and RegionPlot[inequality, vars...] are not equivalent. For instance, try DiscretizeRegion[reg1] and DiscretizeRegion[reg1, Method -> "RegionPlot"]. I think your (failed) RegionPlot uses the first approach. $\endgroup$ – Michael E2 Dec 3 '16 at 22:00
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THIS IS JUST AN EXTENDED COMMENT

The region definition can be simplified to

reg2 = ImplicitRegion[
   1 + x^2 + y^2 <= 10 (x + y),
   {x, y}];

Area[reg2]

(*  49 π  *)

Show[
 RegionPlot[reg2],
 ContourPlot[
  Log10[1 + x^2 + y^2] == 1 + Log10[x + y],
  {x, -2, 12}, {y, -2, 12},
  ContourStyle -> Red,
  PlotPoints -> 50]]

enter image description here

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