What does this PNP transistor + OPAMP circuit does?
This is a standard textbook building block- an antilog amplifier. You'd normally have a diode from the emitter to ground to prevent the input from going too far negative and possibly damaging the transistor (by reverse B-E breakdown). 

How it works:-

The transistor base-collector voltage is maintained at 0V by the op-amp through virtue of negative feedback. 

Collector current is \$i_C = I_S e^{(\frac{V_{BE}}{V_T})}\$, so 

\$Vo = -(100K) I_S e^{(\frac{V_{i}}{V_T})}\$

There is a temperature dependency, obviously in the thermal voltage \$V_T = kT/q\$, but also in the saturation current \$I_S\$, so practical antilog circuits use something like a thermistor to compensate for temperature variations.