To solve this logarithmic equation, we will first use the property that if logₐ(m) = logₐ(n), then m = n.

Given:
Log^3(6+x-x^2) = log3(9+3x-2x^2)

Now, since both sides are equal, we can drop the logarithm and equate the expressions inside them:

6 + x - x² = 9 + 3x - 2x²

Rearrange the terms to set the equation equal to zero:

x² - x + 3 = 0

This is a quadratic equation that we can solve using the quadratic formula:

x = [-(-1) ± √((-1)² - 4(1)(3))] / 2(1)
x = [1 ± √(1 - 12)] / 2
x = [1 ± √(-11)] / 2

Since the square root of a negative number is imaginary, the solutions to this quadratic equation are complex numbers.