#1**+1 **

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**

Guest Jan 6, 2020

#2**+2 **

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.

Daewei Jan 6, 2020

#3**+3 **

2^8 is 256 , so this is equivalent to:

2^{8(x-2)} =2^{(-x-7) Now equate the exponents}

8(x-2) = -x-7

8x-16 = -x-7

9x = 9

x=1

ElectricPavlov Jan 6, 2020