Range of #x - floor(x)# is #[0,1)#
#0# is not in the domain of #ln#, and as #x# gets close to #0#, #lnx# decreases without bound (goes to #-oo#).
#ln1 = 0#, but #x - floor(x)# never takes on the value #1#, so
the range of #ln(x-floor(x))# is #(-oo,0)#