In case you are having trouble reading that:
[itex](a-1)!/((a-1!*0!)) + (a-1)!/((a-2)!*1!) + (a-1)!/((a-3)!*2!)... + (a-1)!/(0!*(a-1)!) = 2^{a-1}[/itex]
Assuming a = positive integer

Essentially, I haven't had much experience with logs, and would be interested in a hint to making this a little more workable.

The next part incorporates part of a logic problem. If you haven't completed (click here), don't continue

Spoiler

Each term represents a number of combinations that occurs with a specific number of objects. For example, if you have 10 people and want to find out how many combinations there are of them going to a party (1,6,10), (1,3,5,9,10) etc; you can find out by finding how many different combinations of 1 person there are, how many combinations of two people, etc. Essentially, I developed a chart to represent it:
1 1 1 1
1 2 3 4
1 3 6 10
1 4 10 20

And a formula to find any term where x=row # and y=column # iii(x+y-2)!
---------------
(x-1)!*(y-1)!
The series above represents a sum of the diagonal of the chart. Someone else also noticed that the same problem could be solved using binary, hence the fact that they equal each other.
I can go into more detail about how I developed this if requested