You want the y axis on your spectrum analyzer to be proportional to log(Vrms), not Vrms, because the input signals have huge dynamic range. They might be 0.000000001Vrms or .001Vrms or 1.0Vrms -- you want it all to fit on the same screen, so you must have that logarithm. The newfangled strategy is to sample a block of IF bandwidth with an ADC, take a FFT, and compute y=log(abs(fft(if_signal))). The log amplifier is the old school analog version of this: an analog IF filter plays the role of a RBW bin in the FFT and the circuit after it wants to take the log. This might mean mapping .000000001Vrms to y=0.0V, .00000001Vrms to y=0.1V, .0000001Vrms to y=0.2V, ..., 1.0VRms to y=0.9V. The "abs" typically happens later in this architecture, so technically those "V" should have been "Vrms" and the log-amplified signals get to keep their phase information for a teeny while longer until it gets discarded in the detector immediately afterwards, but we can conceptually group those together for the purpose of finding the y value for the spectrum analyzer graph. To be precise, though, the log amplifier takes a sin wave with huge dynamic range, say .00000001Vrms to 1.0Vrms, and maps that to a sin wave with, say, 0.1Vrms to 1.0Vrms dynamic range.
Side note: the reason to log-amplify before taking abs in the analog world is that AC is kinder to small signals than DC. At DC, every weird physics effect and its dog wants to murder your signal. If you convert .000000001Vrms to .000000001V and then try to take the log, calibration will be a nightmare. Ask the volt-nuts how easy .000000001V is to measure. However, if you turn .000000001Vrms to 0.1Vrms to 0.1V, you make the signal non-fragile before you move to DC land, and that's a much better strategy.
EDIT: Good point, names on the units were confusing, hopefully a bit less so now.