A few weeks ago, during my Further Pure 2 class, we were looking at using hyperbolic functions to solve equations. And we ended up with something along the lines of (e^x-2)(e^x+3) where the roots were Ln(2) and invalid due to Ln(-3) being a nonsensical answer. Now, I quickly reverted to something I had previously learnt. That being when you sqrt a negative number you used to be told it was an invalid answer, until you were taught complex numbers/roots, where this was no longer the case. And my question is, is there possibly a (not necessarily complex/imaginary), but some sort of other solution to the answer of the ln(-x) invalidity? Where like ln(-x) took a value of say like q lots of ln(x) in a similar fashion to how complex roots work. If not, can someone show me or explain why this cannot be? And if perhaps there is an answer to this, is it therefore feasible to suggest that there is no such thing as invalidity, as you can always create some sort of new concept or imaginary value in of which a solution is valid for, where it previously was not? Cheers Si.