A **log amplifier** is an amplifier for which the output voltage *V*_{out} is *K* times the natural log of the input voltage *V*_{in}. This can be expressed as, resistor on capacitor

V
out
=
K
ln
(
V
in
V
ref
)
where *V*_{ref} is the normalization constant in volts and *K* is the scale factor.

The relationship between the input voltage
V
in
and the output voltage
V
out
is given by:

V
out
=
−
V
T
ln
(
V
in
I
S
R
)
where
I
S
and
V
T
are the saturation current and the thermal voltage of the diode respectively.

A necessary condition for successful operation of a log amplifier is that the input voltage, *V*_{in}, is always positive. This may be ensured by using a rectifier and filter to condition the input signal before applying it to the log amp input. As *V*_{in} is positive, *V*_{out} is obliged to be negative (since the op amp is in the inverting configuration) and is large enough to forward bias the emitter-base junction of the BJT keeping it in the active mode of operation. Now,

V
BE
=
−
V
out
I
C
=
I
SO
(
e
V
BE
V
T
−
1
)
≈
I
SO
e
V
BE
V
T
⇒
V
BE
=
V
T
ln
(
I
C
I
SO
)
where
I
SO
is the saturation current of the emitter-base diode and
V
T
is the thermal voltage. Due to the virtual ground at the op amp differential input,

I
C
=
V
in
R
1
, and

V
out
=
−
V
T
ln
(
V
in
I
SO
R
1
)
The output voltage is expressed as the natural log of the input voltage. Both the saturation current
I
SO
and the thermal voltage
V
T
are temperature dependent, hence, temperature compensating circuits may be required.