Based on the diode equation, one can think of a diode as a current to voltage converter.
If we put an opamp in negative feedback and ground it’s non-inverting terminal, we can generate a current from the input source and resistor at the inverting terminal. This current passes through the diode and a voltage appears across the diode.
Since the current is in terms of the input voltage source and the resistor, using the diode equation, we can find a relationship between the input voltage and the voltage across the diode. This will be:
V_D = V_T*ln( V_in/(R*I_s) )
By subtracting the above equation with the following equation: V_T*ln( (1 volt)/(R*I_s) )
We can cancel out the R/I_s term to simply get
V_D = V_T*ln( V_in )
at the output of the subtractor circuit.
Now as for the V_T term, we can simply set up an amplifier circuit and tune the potentiometer to get a gain that perfectly cancels the thermal voltage term. This is not entirely practical though.
Then we can invert the final answer using an inverting amplifier because the actual output of the first stage was -V_D and not simply V_D.
With all that done we should arrive at
V_out = ln( V_in )
