Why is the function $f$ undefined for $x=1$ when $f(x) = \log_x x$.

Question

The Symbolab graphing calculator tells me that the function $f$ such that :

$$f(x) = \log_x x$$

is undefined for $x=1$.

But suppose that $n = \log_11$

Then $1$ to the $n$th power is equal to $1$ (in virtue of the definition of a logarithm), and it seems to me there is a number $n$ satisfying the condition, namely the number $1$, so the ordered pair $(1,1)$ should belong to the function $f$.

So what did I miss?

Answer 1

You could just as well claim that $\log_1 1=2$, because $1^2=1$.

The function $\log_1 x$ is not well defined, because it is supposed to be the inverse of function $1^x$, but this function isn't invertible, even on its image.

Answer 2

Answer redacted because the question has been changed since I answered.

What is the actual message that the calculator gives you? $f(x) = log_b x$ where $b = 1$ is a valid function only if the domain of $f$ is the set containing a single integer, 1. That's a fine domain but not as common as say, the set of all integers or all real numbers. In those cases, your function would not be a valid function, as there exist some elements of the domain set (eg: 2) that are not mapped to anything in the codomain. It might be that your calculator is interpreting your definition $f$ to be acting on a larger domain than the set just containing 1, in which case, it fails to be a function.