I'm not sure where you learned this, but there is a lot of extraneous math in your solution. All the "1^ -1" is completely unnecessary and confusing, although it does somehow give the correct solution. Here's a simpler way to do it:

256^{x-2} = 2^{-x-7} | 256 is equal to 2^{8}, so you can make the bases the same

2^{8x-16}=2^{-x-7} | Now, you can take away the bases and solve algebraically. If you perform log_{2} on both sides, you're left with just the exponents, so I'm just not showing a step.

8x-16=-x-7 | Next, you solve algebraically.

8x+x~~-16+16~~=~~-x+x~~-7+16

9x=9 | Divide both sides by nine.

x=1 | You have your answer!

Now, let's plug it back in to check if it works.

256^{x-2} = 2^{-x-7} | Replace x with our answer, 1

256^{1-2}=2^{-1-7} | Change 256 to 2^{8}

2^{8-16}=2^{-1-7} | Simplify

2^{-8}=2^{-8}

It works! It may look hard at first, but once you try it out it's much easier and saves you the extra steps.

TLDR; Match your bases, take the log and solve algebraically.