To solve this equation, we need to first rewrite the equation using the properties of logarithms.

Using the property log_a(b) = log_a(c) if and only if b=c, we can rewrite the equation as:

x^2 - x - 2 = 3 - x^2 + 2x

Now, we simplify the equation by combining like terms and rearranging:

x^2 - x - 2 = 3 - x^2 + 2x
2x^2 - 3x - 5 = 0

Now, we have a quadratic equation in the form of ax^2 + bx + c = 0. We can solve this equation by factoring, completing the square, or using the quadratic formula. Let's use the quadratic formula:

x = (-(-3) ± √((-3)^2 - 4(2)(-5))) / (2(2))
x = (3 ± √(9 + 40)) / 4
x = (3 ± √49) / 4
x = (3 ± 7) / 4

Therefore, the solutions to the equation are x = 5/2 or x = -1.