Notation: $\log(\infty)$

Question

In the context of infinite series and indefinite integrals involving $\log(x)$, it's possible to have to substitute $\infty$ into $\log(x)$.

I've seen a paper, where this was marked: $$\log(x)\space |_{1}^{\infty} = \lim_{a\rightarrow\infty}\log(a)-\log(1)$$

The reason for using limits is claimed to be that $\infty$ is not a number.
But why is the limit conception more useful?

No,the times you put $\infty$ as a upper/lower limit (integration, sums, etc) you mean that precise limit at the other side of your equality. — YoTengoUnLCD, Sep 19 '15 at 18:17
@CameronBuie I'm asking for the reason for using the limit notation instead of $\log(\infty)$. — mavavilj, Sep 19 '15 at 18:33

score 1 · Accepted Answer · edited Apr 13 '17 at 12:19

We expect that a number has some properties that $\infty$ has not, e.g. the fact that for $a,b>0 \rightarrow a+b>a$, but this simple properties is not valid if $a=\infty$. So $\infty$ is not a number (but for a better discussion of this topic you can see: Is infinity a number?).

So $\log (\infty)$ has not a meaning as a number, but it can be used as a shortcut for the notation $ \log (\infty)=\lim _{x \rightarrow \infty} \log x$, and this has a precise mathematical meaning thanks to the definition of limit at infinity.

Notation: $\log(\infty)$

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