This is a tough one. I used a tecnique I found by Jonathan Drummey in this post - Logarithmic Scale issue: Bar chart is not appearing in "X" axis for Value 1 to change the calculation for one of the axis to a Log value on the Rows shelf. The log values are better to hide on the axis and instead use the tooltip or show text on the viz to display this info.
I attached a 10.2 version of this viz updated here as well if that helps.
I hope this helps,
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I have also faced the same problem where value of 1 does not show on a logarithmic scale axis.
To overcome this I used a dirty hack of adding an offset of 0.1 - 0.4 where values 1 via means of creating a calculated field.
For aggregated / calculated values like Count, Sum or Average, I created a custom calculated field.
Example of formula used in calculated field: COUNT( [Id] ) + 0.4
The additional value of 0.4 helps the bar to be shown on the bar chart. (offset of 0.4 makes the bar looks more prominent)
Also do not forget to change the number format for the calculated field to have zero decimal places.
By doing this, any value between 0.1 - 0.4 will be rounded down so in terms of presentation of the Mark Label on the bar chart it will show the same value as if the offset was never applied.
This is not perfect but a workaround. Not sure if this will suit your need but I hope it helps.