Due to the non linear behavior of the transistor currents we might get the effect of a logarithmic output, it's scaled with 0.026 due to the thermal voltage :) review ebers moll equations to understand
The formula that gobernates the circuit is:
Vo=ln(vi/r*io + 1) where io is the reverse current of the diode, if we fix r*io=1 and get input voltage minus 1, well get a natural log amplifier
