To solve the equation log2^2(x-1) - 5log2(x-1) - 6 = 0, we can use a substitution to simplify the equation. Let's denote log2(x-1) as y.

Substituting y into the equation, we get:
y^2 - 5y - 6 = 0

Now, we can factor this quadratic equation:
(y - 6)(y + 1) = 0

Setting each factor to zero, we get:
y - 6 = 0
y = 6

and
y + 1 = 0
y = -1

Now, we can substitute back in the original variable:
log2(x-1) = 6
x-1 = 2^6
x-1 = 64
x = 65

and
log2(x-1) = -1
x-1 = 2^-1
x-1 = 1/2
x = 1/2 + 1
x = 3/2

Therefore, the solutions to the equation log2^2(x-1) - 5log2(x-1) - 6 = 0 are x = 65 and x = 3/2.