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In my college calculus class we just covered properties of logs, and I wanted to ask about them. Two of them are these:

For all $0 < a$: $$\log_a1=0$$ and for all $-\infty\leq b \leq \infty$: $$\log_11=b$$

So my question here is about $\log_11$. From the first property, $\log_11=0$ but from the second property, $\log_11=1$. Since 0 does not equal 1, how is this true?

Harish Chandra Rajpoot
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Shelby Longbottom
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    There is no such thing as $\log_1$ and there is no such thing as $\log_0$. These are not allowable bases. – JMoravitz Aug 07 '23 at 12:10
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    We actually have $$\lim_{a\to 1}\log_a(1)=0$$ , but this is not $log_1(1)$ for the same reason why $$\lim_{a\to 0} \frac{0}{a}=0$$ is not $\frac{0}{0}$ – Peter Aug 07 '23 at 12:57

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