Here is a modified version of your example spec using Vega scale type "log" (base 10). The field "logVal" is not needed and omitted for clarity.
Note: For log scale, data values must not be 0 or negative.

Vega spec (v 3.0.8)
------------------------
{
"width": 400,
"height": 300,
"padding": {
"left": 20,
"top": 5,
"right": 20,
"bottom": 5
},
"autosize": {
"type": "pad",
"resize": true
},
"data": [
{
"name": "table",
"format": {
"type": "json",
"parse": "auto"
},
"transform": [
{
"type": "collect",
"sort": {
"field": {
"signal": "valueField"
},
"order": "descending"
}
}
],
"values": [
{
"cc": "United States",
"pt": "A",
"total1": 11,
"groupId1": 1,
"value": 600
},
{
"cc": "Greenland",
"pt": "A",
"total1": 4,
"groupId1": 4,
"value": 900
},
{
"cc": "Mexico",
"pt": "G",
"total1": 2,
"groupId1": 5,
"value": 273
},
{
"cc": "Canada",
"pt": "A",
"total1": 2,
"groupId1": 6,
"value": 30
},
{
"cc": "Chile",
"pt": "A",
"total1": 1,
"groupId1": 7,
"value": 105
},
{
"cc": "Brazil",
"pt": "A",
"total1": 1,
"groupId1": 7,
"value": 540
}
]
}
],
"signals": [
{
"name": "valueField",
"value": "value"
},
{
"name": "labelField",
"value": "cc"
},
{
"name": "minValue",
"value": 10.0
},
{
"name": "maxValue",
"value": 2000.0
}
],
"scales": [
{
"name": "xscale",
"type": "log",
"domain": [{"signal": "minValue"}, {"signal": "maxValue"}],
"range": "width"
},
{
"name": "yscale",
"type": "band",
"domain": {
"data": "table",
"field": {
"signal": "labelField"
}
},
"range": "height",
"padding": 0.2
}
],
"axes": [
{
"orient": "bottom",
"scale": "xscale",
"grid": true,
"encode": {
"grid": {
"update": {
"stroke": {
"value": "lightgray"
},
"strokeDash": {
"value": [10, 10]
}
}
},
"labels": {
"update": {
"fill": {"value": "steelblue"},
"angle": {"value": -45},
"fontSize": {"value": 12},
"align": {"value": "right"},
"baseline": {"value": "top"},
"dx": {"value": 2}
}
}
},
"tickCount": 10,
"zindex": 1
},
{
"orient": "left",
"scale": "yscale",
"ticks": false,
"labelPadding": 10,
"zindex": 1.5
}
],
"marks": [
{
"type": "rect",
"from": {
"data": "table"
},
"encode": {
"enter": {
"y": {
"scale": "yscale",
"field": {
"signal": "labelField"
}
},
"height": {
"scale": "yscale",
"band": 1
},
"x": {
"scale": "xscale",
"field": {
"signal": "valueField"
}
},
"x2": {
"scale": "xscale",
"signal": "minValue"
}
}
}
}
]
}