We can simplify the given equation by applying the properties of logarithms:

Log16 (x^2-2x-3)^2 - 2 log16(x^2+x-2) = 1/2

Using the property of logarithms that states log(a^b) = b*log(a), we can simplify the equation further:

log16((x^2-2x-3)^2) - log16((x^2+x-2)^2) = 1/2

log16((x^2-2x-3)^2 / (x^2+x-2)^2) = 1/2

Now, we can rewrite the equation using the definition of logarithms:

16^(1/2) = (x^2-2x-3)^2 / (x^2+x-2)^2

Solving for the square root:

4 = (x^2-2x-3) / (x^2+x-2)

Now, we solve for x:

4(x^2+x-2) = x^2-2x-3
4x^2 + 4x - 8 = x^2 - 2x - 3
3x^2 + 6x - 5 = 0

Using the quadratic formula, we find:

x = (-6 ± √(6^2 - 43(-5))) / (2*3)
x = (-6 ± √(36 + 60)) / 6
x = (-6 ± √96) / 6
x = (-6 ± 4√6) / 6
x = -1 ± 2√6

Therefore, the solutions to the equation log16 (x^2-2x-3)^2 - 2 log16(x^2+x-2) = 1/2 are x = -1 + 2√6 and x = -1 - 2√6.