To solve this logarithmic equation, we can use the property that if Log(a) = Log(b), then a = b.

From the given equation: Log2(x-2) = Log2(x^2-x-17)
We can set the expressions inside the logarithms equal to each other:
x - 2 = x^2 - x - 17

Now, let's rearrange the equation into standard form to solve for x:
0 = x^2 -x - 17 - (x - 2)
0 = x^2 - 2x - 15

Now we have a quadratic equation that we can solve by factoring or using the quadratic formula:
0 = (x - 5)(x + 3)

Setting each factor to zero gives us:
x - 5 = 0 or x + 3 = 0

Solving for x gives us:
x = 5 or x = -3

Therefore, the solutions to the equation Log2(x-2) = Log2(x^2-x-17) are x = 5 and x = -3.