the logarithmic amplifier
A logarithmic amplifier produces an output proportional to the logarithm of its input, squashing an enormous range of input, say from 1 mV to 10 V, into a small, manageable output range. It does this by exploiting a deep fact of physics: the voltage across a diode or transistor junction is related to the current through it by a logarithm.
Put a diode or, better, a transistor in the feedback path of an inverting amplifier. A pn junction obeys I roughly equals Is times (e to the power V over VT minus 1), which rearranges to V roughly equals VT times the natural log of I over Is. The input voltage Vin pushes a current Vin over R through the junction, so the output becomes roughly minus VT times the natural log of Vin over (R times Is). The practical effect is that every tenfold increase in the input adds a fixed step to the output, on the order of 60 mV per decade times a scaling factor.
Log amps compress wide-dynamic-range signals such as light or audio, and let you compute products and ratios by taking logs, adding, and taking the antilog. The honest caveat is temperature: VT rises with temperature and the saturation current Is roughly doubles every 10 degrees Celsius, so a raw log amp drifts badly. Usable designs add a matched reference transistor and temperature compensation, and they work for only one input polarity.
A photodetector signal that spans six decades of light intensity, a million-to-one range, is hard to display. A log amp compresses it so all six decades fit on a single meter scale.
Taking the log turns a million-to-one range into a tidy, linear-looking scale.
It relies on a junction's exponential law, which drifts hard with temperature. Usable log amps need a matched reference transistor and temperature compensation, and work for only one input polarity.