I understand that $\log_1(1)$ is considered an indeterminate form, but the expression $\log_0(0)$ seems even more subtle. Algebraically, it is undefined because a logarithm cannot have a base of zero, yet from the perspective of calculus and limit analysis, it behaves like an indeterminate form, as the valuecould vary depending on how the base and the argument approach zero. How can one reconcile these two perspectives, and why is it correct to describe $\log_0(0)$ as both undefined and indeterminate at the same time?
4
- 1$\begingroup$ Does something like this link answer your question? math.stackexchange.com/questions/1257027/… (It's not exactly the same thing, but your confusion might just be about the difference between undefined and indeterminate.) $\endgroup$Misha Lavrov– Misha Lavrov2025-10-08 04:48:40 +00:00Commented Oct 8 at 4:48
- $\begingroup$ I understand that an indeterminate form is one whose value cannot be determined directly, like $0/0$, $0^0$, or $\log_1(1)$. I also know that $1/0$, $0^{-1}$, and $log_1(2)$ is undefined. However, i've also seen that in the context of limits according to some claims $\log_0(0)$ is considered to be listed as an indeterminate form while simultaneously being undefined as a value. $\endgroup$Əndəə Demiri– Əndəə Demiri2025-10-08 04:54:45 +00:00Commented Oct 8 at 4:54
- $\begingroup$ FWIW one should be careful to distinguish between "indeterminate" with "indeterminate form". $\frac 00$ is undefined and indeterminate. But say something (rather informally I'm afraid) like "When I try to evaluate $\lim_{x\to 1} \frac {x^2-1}{x-1}$ I get something of the form $\frac 00$. That doesn't mean the limit is $\frac 00$ and undefined and indeterminate. It means that form and evaluting it directly as an indeterminate form and we have to do something else. $\endgroup$fleablood– fleablood2025-10-08 05:36:59 +00:00Commented Oct 8 at 5:36
- $\begingroup$ Is the distinction that you might say that while $0^0$ might indeterminate in terms of different limits, it is widely defined to be $0^0=1$ in combinatorics and related subjects? $\endgroup$Henry– Henry2025-10-08 07:09:09 +00:00Commented Oct 8 at 7:09
Add a comment |
1 Answer
$\begingroup$ $\endgroup$
2 It's fairly common for an expression to be both undefined and indeterminate; the simplest example is probably $ 0 / 0 $, which is undefined (because division by $ 0 $ is usually left undefined) and also indeterminate (because you can't tell the value of a limit just from knowing that it takes this form). But these are really separate questions; one is about evaluating an expression exactly, while the other is about taking limits.
The reason why $ \log _ 1 1 $ is undefined is that there are multiple solutions to the equation $ 1 ^ x = 1 $ and mathematicians have not found it useful to adopt a convention choosing any one of them as the value of the logarithm. The reason why $ \log _ 1 1 $ is indeterminate is that (for example) $ \lim \limits _ { x \to 1 } \log _ x x = 1 $ while $ \lim \limits _ { x \to 1 } \log _ x ( x ^ 2 ) = 2 $, giving two different limits with the form $ \log _ 1 1 $.
The reason why $ \log _ 0 0 $ is undefined is that there are multiple solutions to the equation $ 0 ^ x = 0 $ and mathematicians have not found it useful to adopt a convention choosing any one of them as the value of the logarithm. The reason why $ \log _ 0 0 $ is indeterminate is that (for example) $ \lim \limits _ { x \to 0 ^ + } \log _ x x = 1 $ while $ \lim \limits _ { x \to 0 ^ + } \log _ x { x ^ 2 } = 2 $, giving two different limits with the form $ \log _ 0 0 $.
So for both expressions, both labels are appropriate, but you would use one or the other depending on what you're trying to say: that there is no value to evaluate it to (undefined), or that there is no way to determine the value of a limit taking this form (indeterminate).
- $\begingroup$ Well, $\log_1 x$ is always undefined, not just for $x=1.$ I guess if we could define it for $x=1,$ it could be defined as a function with a one-element domain, but it would hardly be useful to do so. But we do define other one-sided inverse functions, like $\arctan(x),$ by picking one of infinitely many possible $y$ with $x=\tan y.$ $\endgroup$Thomas Andrews– Thomas Andrews2025-10-08 20:28:11 +00:00Commented Oct 8 at 20:28
- $\begingroup$ @ThomasAndrews : Yes, that's why I say‘mathematicians have not found it useful to adopt a convention’ for $\log_11$ or $\log_00$. We could pick a solution to $1^x=1$ or $0^x=0$ but haven't found a compelling reason to. $\endgroup$Toby Bartels– Toby Bartels2025-10-09 04:48:38 +00:00Commented Oct 9 at 4:48