How are we to interpret [the logarithm of one dollar] in the above context of utility? 

You pick an arbitrary constant A of dimension "amount of money", and use log(x/A) as an utility function. Changing A amounts to adding a constant to the utility (and changing the base of the logarithms amounts to multiplying it by a constant), which doesn't affect expected utility maximization. EDIT: And once it's clear that the choice of A is immaterial, you can abuse notation and just write “log(x)”, as Kindly says.