Misconceptions about random numbers in a range I am currently working on a project the includes fitting. For the fitting I would like to try uniform random starting parameters. The possible fit parameters can lie in quite a large range, for example 10^-15 to 10^15 while others can lie in the range of -1 to 1. So quite diverse.
When I try to create random numbers that reflect this range like
Number = Rnd(0, 1) * (UpperLimit - LowerLimit) + LowerLimit

This approach however is pretty useless for the large range. Here pretty much all resulting random numbers are very large. I have yet to see a number smaller than 10^12 for the above range.
For the large range I can reduce the problem by using logarithms:
Number = 10^(Rnd(0, 1) * (Lg(UpperLimit) - Lg(LowerLimit)) + Lg(LowerLimit))

This now produces useful numbers over the whole logarithmic range yet it does not work for negative ranges or zero. 
I think I haven't really understood the basic concepts. If you plot the large numbers in a point plot they seem very uniform, yet they only reflect the large values. Does that mean that small numbers are less likely to turn up? 
The logarithmic numbers of course only look uniform in logarithmic scaling.
I can't be the first person to run unto these problems. How are random numbers usually chosen? In the first step I don't care about quality of randomness. I just want to reflect larger and smaller number ranges including negative and 0 randomly.
If I my explaination was to confusing please leave a comment and I will try to elaborate. As you may have noticed from the question I am note very well educated in statistics. So if you could keep it in Layman's terms I would be very grateful.
Thanks in advance.
 A: 
For the fitting I would like to try uniform random starting parameters

Note the word uniform there. 

This approach however is pretty useless for the large range. Here pretty much all resulting random numbers are very large

This is a consequence of your desire for it to be uniform. If you choose numbers that are uniform on $(10^{-15}, 10^{15})$, then, perforce, about 99.9% of them exceed $10^{12}$.
If that doesn't reflect your desire, then your expressed wish for "uniform random starting parameters" doesn't reflect your actual desire in spite of the fact that you stated it explicitly. The word uniform means something; you must express your wishes with care.

How are random numbers usually chosen? 

That depends on what you're trying to achieve; since your wishes as expressed in the question are self-contradictory, that's something to work out.
As you note, if the value was only positive, you seem to want something more like a uniform over the index (i.e. uniform in the logs), but you say you can't limit yourself to positive numbers. 
You can't do a log scale across 0, of course. How should it behave in the neighborhood of zero? Are you after something that's maybe roughly uniform-ish near zero but more like uniform in the logs (with appropriate change of sign for negatives, presumably) when far from zero? Should it treat negative values (their sign aside) the same as positives in terms of how the scale works (sorry, I expressed that badly)?
