I think your problem is that this statement is not true: "The fact is that likelihood can be in range 0 to 1. "
There's nothing wrong with a positive log likelihood. The likelihood is the product of the density evaluated at the observed values. Probability densities can take "large" positive values depending on how concentrated they are. Don't confuse PDFs with PMFs (probability mass functions). Remember, PDFs integrate to 1, they are not bound by values in the interval [0,1]
For example, consider the N(0,0.1^2) PDF
x = -1:0.01:1;
y = normpdf(x,0,0.1);
Note you can clearly find values of the PDF in the above that result in positive logs.