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As zero value is not supported in logarithmic axis, you can set the value to null by parsing through the datapoints before rendering. Please find the code-snippet below.

function parseZeroValue() {
  var data = chart.options.data;
  for(var i = 0; i < data.length; i++) {
    for(var j =0; j < data[i].dataPoints.length; j++) {
    var yValue = data[i].dataPoints[j].y;
      if(yValue === 0) {
        data[i].dataPoints[j].y = null; //Set this to some value like 0.01, if you like to show column
        data[i].dataPoints[j].toolTipContent = ("{x}: " + yValue);

Please take a look at this JSFiddle for complete code. Also refer to this JSFiddle which shows how to set minimum height in column chart.
Setting minimum height in column chart

Vishwas R
Team CanvasJS