This is a regression in octave-4.4.0.
At least with octave-4.2.1 it was possible to
create a fake axis to attach a standalone colormap
with log scale ticks:

graphics_toolkit("gnuplot")
figure
x = -3:3;
y = x.';
S = exp(-(x.^2 + y.^2));
pcolor(x, y, log10(S))

log_clims = get(gca, "clim");

ax_cb = axes;
set(ax_cb, "visible", "off")
caxis(ax_cb, 10.^(log_clims))
cb = colorbar(ax_cb)
set(cb, "yscale", "log")
set(cb, "yminortick", "on")

But the correct fix is probably to implement the "ColorScale" property:
https://fr.mathworks.com/help/matlab/ref/matlab.graphics.axis.axes-properties.html#budumk7-ColorScale

I have run Rik's test from comment #3 in Matlab. It turned out that newer Matlab releases also do NOT support log-scale colorbars any more!

Therefore I suggest to lower the severity of this bug report to a WISH (for an Octave-only feature) or only do a slight code change to make Octave's behavior Matlab compatible and then CLOSE it.

Here are the details on the Matlab behavior (R2014b):

>> imagesc(eye(100)+1);
h=colorbar()
set (h, 'YScale', 'log')

h =

  ColorBar with properties:

    Location: 'eastoutside'
      Limits: [1 2]
    FontSize: 9
    Position: [0.8194 0.1091 0.0496 0.8155]
       Units: 'normalized'

  Show all properties

Error using matlab.graphics.illustration.ColorBar/set
There is no YScale property on the ColorBar class.

>> get (h)
            AxisLocation: 'out'
        AxisLocationMode: 'auto'
            BeingDeleted: 'off'
                     Box: 'on'
              BusyAction: 'queue'
           ButtonDownFcn: ''
                Children: []
                   Color: [0.1500 0.1500 0.1500]
               CreateFcn: ''
               DeleteFcn: ''
               Direction: 'normal'
               FontAngle: 'normal'
                FontName: 'Helvetica'
                FontSize: 9
              FontWeight: 'normal'
        HandleVisibility: 'on'
                 HitTest: 'on'
           Interruptible: 'off'
                   Label: [1x1 Text]
                  Limits: [1 2]
              LimitsMode: 'auto'
               LineWidth: 0.5000
                Location: 'eastoutside'
                  Parent: [1x1 Figure]
           PickableParts: 'visible'
                Position: [0.8194 0.1091 0.0496 0.8155]
                Selected: 'off'
      SelectionHighlight: 'on'
                     Tag: 'Colorbar'
           TickDirection: 'in'
    TickLabelInterpreter: 'tex'
              TickLabels: {11x1 cell}
          TickLabelsMode: 'auto'
              TickLength: 0.0100
                   Ticks: [1 1.1000 1.2000 1.3000 1.4000 1.5000 1.6000 1.7000 1.8000 1.9000 2]
               TicksMode: 'auto'
                    Type: 'colorbar'
           UIContextMenu: [1x1 ContextMenu]
                   Units: 'normalized'
                UserData: []
                 Visible: 'on'

>>

I will also attach the resulting Matlab figure.

There are also Matlab users complaining about the removal of this feature:

• https://de.mathworks.com/matlabcentral/answers/100066-how-do-i-create-a-logarithmic-scale-colormap-or-colorbar
• https://de.mathworks.com/matlabcentral/answers/203697-logarithmic-scale-for-colorbar

There is still a slight descrepancy between Octave and Matlab in this respect:

• Matlab throws an error when you try to create a log-scale colorbar. The colorbar stays colored (with a lin-scale).
• Octave just displays the "log-scale" colorbar but silently takes all the colors out.

Maybe we should also just throw an error and keep the colors lin-scale?

@Thomas: Can you run the test code in Matlab and upload what the log-scale colorbar image should look like?

The test code was:

imagesc(eye(100)+1);
h=colorbar()
set (h, 'YScale', 'log')

AFAIK colorbars are built with an image. Non linearly spaced image pixels are not supported neither in octave nor in matlab.
 
A workaround may be to use a flat surface as proposed here [1], to build a custom colorbar. E.g:

imagesc (eye (100) + 1)
h = colorbar ();
hi = get (h, "children");
dat = get (hi, {"xdata", "ydata", "cdata"});
z = [dat{3} dat{3}];
x = dat{1};
y = linspace (dat{2}(1), dat{2}(2), rows (z));
delete (hi)
surface (h, x, y, z, "edgecolor", "none");
axis (h, 'tight')
set (h, 'yscale', 'log')

[1] http://www.mathworks.com/matlabcentral/newsreader/view_thread/321116