Solve for x :
256^(x - 2) = 2^(-x - 7)
-256^(x - 2) = (-1)^1·2^(8 x - 16):
(-1)^1·2^(-x - 7) = (-1)^1·2^(8 x - 16)
Equate exponents of -1 and 2 on both sides:
1 = 1 and -x - 7 = 8 x - 16
All equations give x = 1 as the solution:
x = 1
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:
256x-2 = 2-x-7 | 256 is equal to 28, so you can make the bases the same
28x-16=2-x-7 | Now, you can take away the bases and solve algebraically. If you perform log2 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.
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.
256x-2 = 2-x-7 | Replace x with our answer, 1
2561-2=2-1-7 | Change 256 to 28
28-16=2-1-7 | Simplify
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.