I want to know why log(9x^2) = 0 has two solutions. If we write it as log(9) + log(x^2) = 0, ==> log(9) + 2 log(x) = 0 ==> 2log(x) = -log(9) ==> log(x) =-[log(9)]/2 ==> x = 10 ^ ([-log(9)]/2), there is nothing in the algebraic manipulation that would suggest two solutions because each move was via some rule of common logs. Of course, i understand intuitively why it has two solutions, but why does the algebraic manipulation allow us to arrive at only one solution?