# Log expression gives two answers based on how you think about it

when you have the expression $$\log( -x)$$ base $$3$$, isn't that undefined, because you can't raise 3 to any power which will give you a negative value?

I am supposed to graph this expression and from the transformation of functions, this makes sense though because you just flip the graph of $$log x$$ base $$3$$ over the y-axis.

I would like to know why you get two different answers?

Thanks, Amruth

• What makes you think that $-x$ is negative? Jan 11 at 0:44
• $-x$ could be positive if $x$ is negative! Jan 11 at 0:58

Since $$\log(x)$$ is only defined on $$(0,\infty)$$ we get that $$\log(-x)$$ is only defined on $$(-\infty, 0)$$ Morever, given $$f(x) = \log(x)$$, we get that $$f(-x) = \log(-x)$$ is the reflection about the y-axis.
Hence, we do not get two answers from the same input of the given function. That is to say : $$\log$$ is a well-defined function.
Also see here : What is $\text{log}(-x)$?