I am probably just missing something.

Solve:  2^x-1 + 2^x-4 + 2^x-2  =  6,5555...

First:

     6,55555...  =  6 + 5/9  =  54/9 + 5/9  =  59/9

The terms:  2^x-1,  2^x-4, and 2^x-2 can all be written in terms of 2^x-4:

     2^x-1  =  2^x-4+3  =  2^x-4·2³  =  2^x-4·8

     2^x-4  =  2^x-4·1

     2^x-2  =  2^x-4+2  =  2^x-4·22  =  2^x-4·4

Therefore, the problem can be re-written as:

     2^x-4·8 + 2^x-4·1 + 2^x-4·4  =  59/9

Factoring out the term  2^x-4:

     2^x-4(8 + 1 + 4)  =  59/9

     2^x-4(13)  =  59/9

Dividing by 13:

     2^x-4  =  (59/9) / 13

     2^x-4  =  59/117

Taking the log of both sides:

     log( 2^x-4 )  = log(59/117)

Simplifying:

     (x-4)·log(2)  =  log(59/117)

Dividing by log(2):

     x - 4  =  log(59/117) / log(2)

Adding 4:

     x  =  log(59/117) / log(2) + 4

     x  =  3.012278... 

Solve for x over the real numbers:
2^(x - 4) + 2^(x - 2) + 2^(x - 1) = 59/9

Simplify and substitute y = 2^x.
2^(x - 4) + 2^(x - 2) + 2^(x - 1) = (13×2^x)/16
 = (13 y)/16:
(13 y)/16 = 59/9

Multiply both sides by 16/13:
y = 944/117

Substitute back for y = 2^x:
2^x = 944/117

Take the logarithm base 2 of both sides:
x = (ln(944/117))/(ln(2))

This is the problem:

What number does X have to be so that the following equasion is correct:

2^(x-1) + 2^(x-4) + 2^(x-2) = 6,5555...