Differential pairs have a hyperbolic tangent response for voltage-in current-out. So these are often used with a cascade of gain stages to do the log approximation over a very wide dynamic range. See the attached figure 11 from HP Journal article about the old 8757 scalar network analyzer.
Another approach is make a variable-in-dB gain amplifier using a resistor ladder attenuator, and putting this in a feedback loop to do a log detector. This is described in attached patent US5077541. It has lots of schematics for the brilliant idea, authored by Barrie Gilbert. This is used in some of the Analog Devices log detectors.