Combining two plots with different x-axes and not labeling the x-axis

This can be done with Overlay. A few useful tricks for best results are

1. Specify the same Frame option in both plots. This determines which ticks and frame labels show.
2. Specify the sameImageSize in both plots.
3. Specify the same ImagePadding in both plots. This aligns the axes and allows room for the frame labels.
4. Use GridLines in one plot, but not in the other
5. Use the PlotRange option to force the same or different scales.

Here is an example that applies the above tricks.

plt1 = ListLogLogPlot[{{10^16, 1000}, {10^19, 900}, {10^22, 1}},
   GridLines -> Automatic,
   PlotStyle -> Magenta, Joined -> True,
   PlotRange -> {All, {1/10, 2000}},
   ImageSize -> 250, ImagePadding -> {{50, 5}, {5, 5}},
   Frame -> {{True, False}, {False, False}},
   FrameLabel -> {"Carrier Density (1/!(*SuperscriptBox[(cm), 
(3)]))", "!(*SubscriptBox[(L), (k, 3  D)]) (nH/[Mu]m)"}];
plt2 = ListLogLogPlot[{{10^4, 3}, {10^7, 3}, {10^8, 1}},
   PlotStyle -> Blue, Joined -> True,
   PlotRange -> {All, {1/10, 2000}},
   ImageSize -> 250, ImagePadding -> {{50, 5}, {5, 5}},
   Frame -> {{True, False}, {False, False}},
   FrameLabel -> {"Unknown quantity", None}];
Overlay[{plt1, plt2}]

You can use ResourceFunction["PlotGrid"] to do this:

plt1 = ListLogLogPlot[
  {{10^16, 1000}, {10^19, 900}, {10^22, 1}},
  Frame -> True,
  Joined -> True,
  FrameLabel -> {"Carrier Density (1/!(*SuperscriptBox[(cm), (3)]))", "!(*SubscriptBox[(L), (k, 3  D)]) (nH/[Mu]m)"}
  ]
plt2 = ListLinePlot[
  {{10^4, 3}, {10^7, 3}, {10^8, 1}},
  Frame -> True,
  FrameLabel -> {"Unknown quantity", None}
  ]

ResourceFunction["PlotGrid"][{{plt1}, {plt2}}]

PlotGrid will simply stack the plots exactly on top of each other, even if the plot ranges are different (note the mismatched ticks in between the plots.

An alternative solution is to use ResourceFunction["CombinePlots"] with the "AxesSides" options:

plt1 = ListLogLogPlot[
   {{10^16, 1000}, {10^19, 900}, {10^22, 1}},
   Frame -> True,
   PlotStyle -> Blue,
   FrameStyle -> Blue,
   Joined -> True,
   FrameLabel -> {"Carrier Density (1/!(*SuperscriptBox[(cm), (3)]))", "!(*SubscriptBox[(L), (k, 3  D)]) (nH/[Mu]m)"}
   ];
plt2 = ListLinePlot[
   {{10^4, 3}, {10^7, 3}, {10^8, 1}},
   Frame -> True,
   PlotStyle -> Red,
   FrameStyle -> Red,
   FrameLabel -> {"Unknown quantity", None}
   ];

ResourceFunction["CombinePlots"][plt1, plt2, "AxesSides" -> "TwoXY"]

Note how I styled both the lines and the frame of the two plots differently to make it easier to tell which axis belongs to which line. This has the added advantage of showing both lines directly on top of each other, while still preserving all the plot range information.