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How can I create a smooth histogram with a log-log scale?

I can use Histogram[data, "Log", "LogCount"] to get a log-log histogram, or I can use SmoothHistogram[data] to get a smooth histogram, but is there a way to combine these two functionalities?

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You can simply get the SmoothKernelDistribution and build the plot as you'd like:

data = Table[Sin[x]^3 + 1, {x, 0, 6 Pi, 0.1}];
dist = SmoothKernelDistribution[data];
LogLogPlot[PDF[dist, x], {x, 0.01, 2}]
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Perhaps an approximation like this may help

Some data from the help:

sizes = FileByteCount /@  FileNames["*.nb", 
        FileNameJoin[{$InstallationDirectory, "Documentation", "English", 
        "System", "ReferencePages", "Symbols"}]];

Show[Histogram[sizes, "Log", "LogCount", Frame -> True], 
     ListLogLogPlot[Transpose[{Rest@#[[1]], #[[2]]} &@HistogramList[sizes, "Log", "LogCount"]], 
                    Joined -> True, InterpolationOrder -> 3, PlotStyle -> Red]]

Mathematica graphics

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Combination of the answers of jVincent and belisarius:

dist = SmoothKernelDistribution[Log[sizes], 0.1];
Show[Histogram[sizes, "Log", "LogPDF", Frame -> True, PlotRange -> All], 
 LogLogPlot[PDF[dist, Log[x]]/x, {x, 10^3, 10^9}, 
  PlotRange -> 10^{-11, -4}, PlotStyle -> Red]]

enter image description here

It uses a smooth kernel with the uniform bandwidth 0.1 in the log scale.

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Starting in M11, you can just give SmoothHistogram the ScalingFunctions option:

data = Table[Sin[x]^3+1, {x, 0, 6 Pi, 0.1}];
SmoothHistogram[data, ScalingFunctions->{"Log","Log"}]

enter image description here

In M10, you can also use the ScalingFunctions option, but you will in addition need to specify the axes origin in transformed coordinates, e.g. (in M10.3.1):

data=Table[Sin[x]^3+1,{x,0,6 Pi,0.1}];
SmoothHistogram[data, ScalingFunctions->{"Log","Log"}, AxesOrigin->Log@{.002, .002}]

enter image description here

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