The starting equation

    (ln x)^3 = ln (x^4)      (1)

implies

    (ln(x))^3 = 4*ln(x),   or

     (ln(x))^3 - 4*ln(x) = 0.


Factor the left side

    ln(x) * ( (ln(x)^2 - 4) = 0;


factor it further

     ln(x)* (ln(x)-2) * (ln(x) +2) = 0.     (2)


Now from  ln(x) = 0,  ln(x) - 2 = 0  and  ln(x) + 2 = 0  for each of the three factors of (2),


you get the three listed solutions.