Why are all log amp examples with common base logging transistors?
The common collector approach has the issue that the base current flows into the input of the opamp through the transistor and causes an error that depends upon the hfe of the transistor.  Also as shown it will not function - you need to put the collector to a negative voltage.

The common emitter design can be used as in the AD8304 (Datasheet).

The reference voltage to the positive input of the opamp cannot be at zero volts though.  In the AD8304 it is raised to about 0.5V to allow operation of the logging transistor. If the input current is from a photodetector this is usually not an issue.

As George points out in his answer a diode connected transistor can also be used fairly successfully. They work much better than ordinary diodes and avoid the excessive gain within the feedback loop of the normal common base approach.

This large gain in the loop creates difficulties in maintaining stability that in turn causes the settling time to be extremely slow at low light levels (I had one design that would work down to -90dBm but take more than a minute to settle).