+1...
First let em get this out of the way.... "why". :
OK on to solutions.
I have designed many log conversions using transistor arrays (the Vbe junction voltage characteristic with current is logarithmic). Nowadays with cheap transistor arrays history, the simplest (accurate) solution is to use a pair of THAT corp analog engines that contain precision rectifiers and log conversions inside ICs (ignore that you would be discarding decent VCAs). Subtract the output log result from input for dB difference (voltage will need to be scaled.) This requires two analog engines so not cheap.
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For a crude GR meter, over a limited dynamic range and resolution, here is an idea to chew on. Using several LM339 quad comparators calibrated to respond to something like 3 dB level steps (over modest range), with an input set of comparators responding to input level, and an output set responding to output level, the 339 comparators could be configured to either add X mA per 3dB of input level, or subtract the same X mA per 3dB of output level from a single summed current node. The net current would represent the level "difference", albeit crudely and over limited dynamic range. This is probably impractical to realize both fine resolution a wide dynamic range but there is no reason why you can't scale the threshold width and output current contribution to provide more resolution around 0VU and less resolution at lower levels (imagine 1dB steps for top 10 dB, 3 dB steps below, 3dB steps provide 3x current per threshold, to maintain scaled results.) This could be reasonably accurate.