Expression about the output voltage Op amp BJT
Despite my (and other commentors') skepticism about this circuit, if you believe LTSpice, this circuit actually does work. Here I've tried both the as-drawn circuit and swapping the emitter and collector pins of the BJT.





With the BJT reversed (input to the emitter), the gain is substantially increased.

I'd expect this to also give a closer fit to an ideal exponential response, but haven't checked it.

In some other testing I found that the 2N3904 model in LTSpice will pull \$V_{ce}\$ to just 10-50 mV if strongly saturated, much lower than the ~0.2 V we often assume for saturation behavior. The model has essentially the same \$V_{ce}(sat)\$ for both forward and reverse operation. I don't know if this is realistic or a model inaccuracy.

(Also, from the charts given, you can find by inspection which of the 4 suggested answers is closest to the simulator result)

Now, how to actually solve the problem

Consider this circuit:



In this circuit, to maintain the two op-amp inputs at the same voltage, the current through R1 must be equal to \$I_1\$, and therefore

$$V_{out} = -R_1 I_1$$

The circuit you're asked to analyze essentially is an initial nonlinear voltage to current converter, followed by this linear current to voltage converter.

So if you can work out what will be the voltage-to-current transfer function of the transistor circuit on the input, you can work out the overall transfer function by multiplying by \$-R_1\$.