How can I have a region plot with the logarithmic axis? In the following, I have brought the original case which I want to be logarithmic. In this case, I have a density plot and a region plot that shows the realm of validity of the theory. I want to show this figure in logarithmic scales in order to be more clear.

\[Eta]4=0.123663

 con14=-0.1 + 8 E^8 (-((5 \[Mu])/(3 E^(40/3))) + (
 8 \[Pi] \[Eta] (BesselI[0, 4] BesselK[0, 4] - 
 BesselI[1, 4] BesselK[1, 4]))/E^8)
 con24=-12.4276 + \[Mu]
 con34=-((5 \[Mu])/(3 E^(40/3))) + (
 8 \[Pi] \[Eta] (BesselI[0, 4] BesselK[0, 4] - 
 BesselI[1, 4] BesselK[1, 4]))/E^8
 con44=(35 \[Mu])/(36 E^(16/3)) + 
 4 \[Pi] \[Eta] ((2 BesselI[0, 4] + 4 BesselI[1, 4]) BesselK[0, 
 4] - (4 BesselI[0, 4] + BesselI[1, 4]) BesselK[1, 4])

 pnrhs4=(1/(\[Eta] Sqrt[
 161.08 \[Eta] + 
 35. \[Mu]]))(-22.1002 \[Eta]^(3/2) Sqrt[67.0519 \[Eta] - 5. \[Mu]]
 Sqrt[\[Mu]] + 
 2.78971 \[Mu]^(
 5/2) + \[Eta]^2 (-255.832 Sqrt[\[Mu]] - 
 0.764016 Sqrt[
 161.08 \[Eta] + 35. \[Mu]]) + \[Eta] (-240.094 \[Mu]^(3/2) + 
 1. Sqrt[161.08 \[Eta] + 35. \[Mu]] + 
 0.00911946 \[Mu] Sqrt[161.08 \[Eta] + 35. \[Mu]]))

 CPN4=(1/(\[Eta] Sqrt[
 161.08 \[Eta] + 
 35. \[Mu]]))(-22.1002 \[Eta]^(3/2) Sqrt[67.0519 \[Eta] - 5. \[Mu]]
 Sqrt[\[Mu]] + 
 2.78971 \[Mu]^(
 5/2) + \[Eta]^2 (-255.832 Sqrt[\[Mu]] - 
 0.764016 Sqrt[
 161.08 \[Eta] + 35. \[Mu]]) + \[Eta] (-240.094 \[Mu]^(3/2) + 
 0.00911946 \[Mu] Sqrt[161.08 \[Eta] + 35. \[Mu]]))

 QNN4=(60.1862 Sqrt[0.0216024 \[Eta] + 0.00469384 \[Mu]] Sqrt[\[Mu]])/\[Eta]

 Area4 = RegionPlot[{con14 < 0 && con24 < 0 && con34 > 0 && con44 > 0 &&
 Abs[CPN4] < 1 && QNN4 > 0}, {\[Mu], 0, 0.028}, {\[Eta], 
 0, \[Eta]4}, BoundaryStyle -> {Green}, PlotStyle -> {None}, 
 FrameLabel -> Automatic]

 P4 = Show[
 DensityPlot[(QNN4/pnrhs4), {\[Mu], 0, 0.028}, {\[Eta], 0, \[Eta]4}, 
 PlotRange -> {0, 75}, PlotPoints -> 300, 
 PlotRangePadding -> {Automatic, 0.00015}, 
 FrameLabel -> {Style[\[Mu], FontSize -> 14, Blue], 
 Style[\[Eta], FontSize -> 14, Blue]}, 
 BaseStyle -> {FontWeight -> Bold, FontSize -> 17}, 
 ColorFunction -> "SunsetColors", 
 PlotLegends -> 
 BarLegend[Automatic, LegendMarkerSize -> 230, LegendMargins -> 5, 
 LegendLabel -> Style["Q/(1+C)", FontSize -> 16], 
 LabelStyle -> {Bold, FontSize -> 14}], 
 FrameTicks -> {{{0, 0.04, 0.08, 0.12}, 
 None}, {{0, 0.006, 0.012, 0.018, 0.024}, None}}], Area4]

Thank you for your help.