real analysis - What is $lim_{x to 0} log_0(x)$?

As per the title; what is $\lim_{x \to 0} \log_0(x)$ ?

According to WolframAlpha: $$\lim_{x \to 0} \log_0(x) = 0$$ but how is this possible?

Surely the limit should be indeterminate since $\log_0(x) = \frac{\log(x)}{\log(0)}$ and $\log(0) =$ indeterminate?

Answer

$\lim\limits_{x\to0}\log_0 x$ cannot exist unless $\log_0 x$ exists for $x$ in some open neighborhood of $0$, with the possible exception of $x=0$. (Since the base must be positive, we can take "open neighborhood of $0$" to mean a set of the form $[0,\varepsilon)$ where $\varepsilon>0$.)

Later edit: I suspect what's going on is something like this:
$$\log_\varepsilon x = \frac{\log x}{\log\varepsilon},$$
then letting $\varepsilon\downarrow0$, we have the denominator going to $-\infty$, so that the fraction approaches $0$. Hence $\log_0 x$ get construed to be $0$, and then one lets $x$ go to $0$, and the limit is $0$.