How to convert exponential output to linear output
First, your 0-6 V range is impossible for taking the logarithm of.  The input can't go to 0, and there will be some maximum ratio of max/min input that any real system can handle.

However, there are such things as log amps that do what you are basically asking about.

You can make a crude log amp yourself by noting that the voltage across a PN junction is roughly the log of the current over a reasonable range.  Therefore, have your input signal control a current source.  This is applied to a diode.  The voltage across the diode is the output.

This is the basis for how log amps work inside, but there will be other compensating circuitry in a real log amp that makes the transfer function more accurate, not so dependent on temperature, etc.