Traditional active filters process signals as voltages, and their dynamic range is bounded on top by the op-amp rails and on the bottom by noise. Log-domain filters escape this trap by compressing the signal into a logarithmic current before filtering, doing the filter math in the log domain, then expanding back to a linear voltage or current at the output. The result: dynamic ranges of 100+ dB from a supply as low as 1.5 V, and cutoff frequencies that tune linearly with a bias current across four or five decades.

The magic exploits the same BJT exponential I_C = I_S·exp(V_BE/V_T) relationship that powers translinear circuits. A signal current is forced through a diode-connected transistor, producing a logarithmic voltage. That log voltage drives the base of another transistor whose emitter connects to a capacitor. Because current into the capacitor sets voltage across it, but that voltage is logarithmically related to the output current, the differential equation governing the output current becomes linear — a first-order lowpass, even though every node voltage is highly nonlinear.

Frey's 1993 log-domain lowpass is the canonical example: four transistors and one capacitor implement a lossy integrator whose −3 dB corner is:

• f_c = I_bias / (2π · C · V_T)

With C = 100 pF, V_T = 26 mV at room temperature, and I_bias swept from 100 nA to 1 mA, f_c ranges from about 6 kHz to 60 MHz — four decades of electronic tuning with a single DC current. No switched capacitors, no clocked artifacts, no aliasing.

Real-world example: Cochlear implant processors use log-domain bandpass filter banks to split incoming audio into 16–22 frequency channels for direct electrode stimulation. The compressed internal signals mean the whole filter bank runs from a 1 V battery drawing microamps, critical for a device sealed inside the skull. Philips and later Cochlear Ltd used log-domain filters extensively in the Nucleus series because linear-mode Sallen-Key filters simply couldn't meet the power budget.

The catch: transistor matching. Log-domain circuits demand tight β and I_S matching — mismatches cause distortion and gain errors that don't cancel like they do in translinear multipliers. This is why log-domain filters are almost exclusively an IC design technique. Discrete versions using matched transistor arrays (LM3046, THAT300) work but rarely justify the effort versus a well-designed OTA-C or gm-C filter. On-chip, though, they're a superpower: high dynamic range, low voltage, wide tuning, all from a handful of BJTs and one capacitor per pole.