Vo = -Rf*Io*e^(Vin/nVt)
Vo - Output voltage
Rf - Feedback resistance = 1Kohm
Io = leakage current of diode
e - exponential
Vin - Input voltage
n - Intrinsic semiconductor rate
Vt - Thermal equivalent of voltage = 26mV
Vo is proportional to e^(Vin/nVt)
Output is proportional to exponent of input. And exponent function is basically antilog.
Therefore the name Antilogarithmic Amplifier