Don't do it: a log transform of the horizontal axis is fine but a log transform (or any transform for that matter) of the vertical axis destroys much of any interpretation of an estimate of a probability density function. Determining differences among different distributions becomes next to impossible.
For example consider samples from Exponential distributions with parameters 1 and 2. Here are the "vertical log-scaled" histograms with the same vertical log scaling:
SeedRandom[12345];
data1 = RandomVariate[ExponentialDistribution[1], 10^6];
data2 = RandomVariate[ExponentialDistribution[2], 10^6];
(* Set binwidth and frequency minimum *)
binwidth = 0.1;
fmin = 1/(binwidth*Length[data]); (* Arbitrary lower bound *)
h1 = HistogramList[data1, {0, 15, binwidth}, "PDF"];
h2 = HistogramList[data2, {0, 15, binwidth}, "PDF"];
(* Replace values of 0 density with fmin *)
h1[[2]] = Max[fmin, #] & /@ h1[[2]];
h2[[2]] = Max[fmin, #] & /@ h2[[2]];
(* Generate rectangles to mimic a histogram *)
t1 = Table[{{h1[[1, i]], fmin}, {h1[[1, i]],
h1[[2, i]]}, {h1[[1, i + 1]], h1[[2, i]]}, {h1[[1, i + 1]],
fmin}}, {i, Length[h1[[2]]]}];
t2 = Table[{{h2[[1, i]], fmin}, {h2[[1, i]],
h2[[2, i]]}, {h2[[1, i + 1]], h2[[2, i]]}, {h2[[1, i + 1]],
fmin}}, {i, Length[h2[[2]]]}];
(* Show results *)
GraphicsRow[{ListLinePlot[t1, PlotRange -> {All, {xmin, Max[h[[1]]]}},
PlotStyle -> Orange, AxesOrigin -> {0, xmin},
ScalingFunctions -> {None, "Log"}, Filling -> Axis,
FillingStyle -> Orange],
ListLinePlot[t2, PlotRange -> {All, {xmin, Max[h[[1]]]}},
PlotStyle -> Orange, AxesOrigin -> {0, xmin},
ScalingFunctions -> {None, "Log"}, Filling -> Axis,
FillingStyle -> Orange]}]
Yes, the two figures appear different but can one visually estimate any characteristic of the distributions? The mean? Any percentile? I don't think so.
Consider histograms without log-scaling:
GraphicsRow[{Histogram[data1, "FreedmanDiaconis", "PDF", PlotRange -> {{0, 5}, {0, 2}}],
Histogram[data2, "FreedmanDiaconis", "PDF", PlotRange -> {{0, 5}, {0, 2}}]}]

One can immediately see differences in means and spread. For such distributions maybe constructing a histogram with the log of the data might be more informative and in that case two SmoothHistograms
rather than standard histograms would be better:
Show[SmoothHistogram[Log[data1], Automatic, "PDF",
PlotLegends -> LineLegend[{Blue, Orange}, {"Exponential[1]", "Exponential[2]"}]],
SmoothHistogram[Log[data2], Automatic, "PDF", PlotStyle -> Orange]]

So even if you got the desired "base", the ability to compared different sets of data is destroyed.
13.0.1 for Linux x86 (64-bit) (January 29, 2022)
I see the same plot of the OP $\endgroup$