I am having some confusion in regards to the log based value of a negative number. I know that this is said to be undefined, though I accidentally entered in '$\log(-x)$' instead of '$\log(x)$' via a graphing application, and it actually graphed a series of $x$-$y$ values, but, once again, I thought this was undefined.
So, like, apparently the value of $\log(-10) = 1$?
I get that you can have negative $y$ values, thus transforming the $y$ values over the $x$ axis, as well as things like $1/\log(x)$, etc., but not $\log(-x)$; I am not really sure what the $-$ sign actually implies (in this context).
By the way, if I enter in $\log(-10)$ in the graphing application of any other mathematical tool, it does indeed say that it is undefined, as expected. I also tested this in other graphing apps, and $\log(-x)$ is for some reason also a legit operation, so it doesn't seem to be some sort of bug in the software.
Thanks to those who answered, it is very much appreciated 🙂
In addition I'd like to mention that I realised that one could use algebra to figure out why -x works. For instance, log_2(-x) = 3 if x=8, thus -x =8, so x must equal -8, as –8 = +8.
Thus flipping the values over the y axis, from 8, to -8, but keeping the value of the y constant. So all positive values of x must be negative, since the negative of a negative is positive.
Best Answer
In the field of real numbers the function $\log$ is defined ( i.e. has real value) only if its argument is positive. So $\log x$ is defined only if $x>0$ and $\log (-x)$ is defined only if $x<0$ and $\log |x|$ is defined only if $x\ne 0$.
We can define the logarithm of negative arguments in the field of complex numbers, as inverse of complex exponentiation, but the definition require some care because the complex exponential is a periodic function, so it is not invertible in a simple manner. You can see here for an introduction.