Electronic – How to make a log amplifier’s logarithmic region span across the range of 0-5V input instead just working close to 0V

What you need is to is to either make the voltage difference between two nodes zero or make open circuiting so that they can not reference each other in order to actually power off.

Cutting one would be open circuiting the internal circuit therefore powering off the op-amp.

As for the "best way": the rest of the circuit is of importance here, since probably powering the op-amp off change other stuff.

Your 2nd circuits (with the NPN and R divider) is a good approximation of a log circuit. This is because when the input opamp generates a voltage substantially above 0.6 V, the current in the 4.7k is proportional to the voltage, and so the V across the NPN is proportional to the log of that current. The 100k & 10k give a multiplier effect, so in fact, your transfer function is closer to VOUT = K*26mVln(Iin/Is), where Iin = (VIN-0.7)/4.7k. 'Is' is hard to find directly, but if you measure (guess) VBE at 1 mA (say 0.6 V), the equation can be rewritten as VOUT = K[26mV*ln(In) + 0.6], where 'In' is in mA.

K is the gain from your R divider -- with 'Contour' =0, it's 1; with 'contour' = 20k, it's 3.

Note that this circuit will change over temperature --if the NPN heats up by (say) 30 deg. C, that's equivalent to about a 10 dB reduction in volume (when you go through math).