Logarithmic scale for large distances?
Having some experience with the only logarithmic unit in regular use (the dB, for sound intensity), I have to say: please, no. 

A logarithmic representation can be handy in some cases. This chart is an excellent example of what can be done if you choose the scale on your image correctly. 



Note that this has been done without introducing a new unit for distance, it just plots log(x) instead of x on the vertical scale. 

But if you were to use a logarithm in the fundamental unit for distance, you'd introduce some problems: 


cosmological distances no longer match other distance units and any conversion between them involves using a log function. 
calculations involving cosmological distances get more complicated because you've now got to use log() everywhere. 
you lose the linear relationship between numbers: object X is twice as far away as object Y, but the difference in your unit becomes log(2) making distances far less intuitive.