Working of transistor in an Opamp log amplifier..?
If you short collector to base and use it as a diode between the opamp output and the inverting input then any base current has to flow from the input signal causing an error, especially at low currents where the HFE drops off.

In the arrangement shown the base current flows to ground and so the current flowing into the input is slightly less. This could cause an error of a few % at low currents.

The configuration shown can provide a logarithmic function over about 9 decades, a diode-connected version can work for 5-6 decades.

Here is some good information that has a brief mention of both types  Log Amplifiers