To solve this logarithmic equation, we will first combine the logarithmic terms using logarithmic rules and properties.

We can rewrite the equation as:

log0.5 (x+2) - log2 (x-3) = (1/2) log1/2 (-4x-8)

Using the property that loga(x) - loga(y) = loga(x/y), we can combine the logarithms on the left side:

log0.5 ((x+2)/(x-3)) = (1/2) log1/2 (-4x-8)

Now, we can rewrite the equation using the change of base formula for logarithms:

log ((x+2)/(x-3))/log(0.5) = (1/2) (log (-4x-8))/log(1/2)

Simplify the equation further:

log2((x+2)/(x-3)) = (1/2) log1/2 (-4x-8)

Now, we can eliminate the logarithms by raising both sides to the power of 2:

2^(log2((x+2)/(x-3))) = 2^((1/2) log1/2 (-4x-8))

Simplify to:

((x+2)/(x-3)) = √(-4x-8)

Now, square both sides and solve for x:

(x+2)^2 = -4x-8

Expanding and simplifying the left side:

x^2 + 4x + 4 = -4x - 8

Rearranging terms:

x^2 + 8x + 12 = 0

Now, we have a quadratic equation. We can solve for x using the quadratic formula:

x = [-8 ± sqrt((8)^2 - 4112)] / 2

x = [-8 ± sqrt(64 - 48)] / 2

x = [-8 ± sqrt(16)] / 2

x = (-8 ± 4) / 2

This gives two possible solutions:

x1 = (-8 + 4) / 2 = -2
x2 = (-8 - 4) / 2 = -6

Therefore, the solutions to the equation are x = -2 and x = -6.