You can more easily check the accuracy of log by using floats that are 
representable exactly in binary.  like
1, 0.5, 0.25, ... if that is your objective.

0.1 read in and converted to a nearby binary double float number is 
approximately 

1.000000000000000055511151231257827021181583404541e-1

For your information,  in Mathematica one can get extra precision by adding 
enough zeros (more than 16 or so)
and in my opinion the design of the significance arithmetic there is deeply 
flawed.  I hope that Sage and
sympy do not take Mathematica as a guide for what the user should see.

RJF


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