I am confused as to what you are saying.
"Ok so ill give an example, x'(t) = log(3t(x(t)-2)) is solution to equation where t0 = 3 and x0 = 5".
x'(t)= ... is NOT a "solution" to a differential equation, it is a differential equation.
Or do you mean to say that x(t)= log(4t(x(t)- 2)) is a solution to some unknown differential equation? Then whether or not that equation is unique depends upon exactly what the differential equation is.