Exp and Log Amplifiers

Recall the known relationship between the current through and voltage across a diode:

$$I_D=I_0 \left( e^{V_D/V_T}-1 \right)$$ (11)

where $I_0$ is the reverse seturation current and $V_T$ is the thermal equivalent voltage. Based on virtual ground and applying KCL to the inverting input of the op-amp, we get

$$\frac{v_{out}}{R}=-I_0(e^{v_{in}/V_T}-1)\approx -I_0\;e^{v_{in}/V_T}, \;\;\;\;\;\; \;\;\;\;\; \frac{v_{in}}{R}=I_0(e^{-v_{out}/V_T}-1)\approx I_0\,e^{-v_{out}/V_T}$$ (12)

i.e.,

$$v_{out}\approx -R I_0 e^{v_{in}/V_T},\;\;\;\;\;\; \;\;\;\;\; v_{out}\approx -V_T\ln(v_{in}/I_0R)$$ (13)