Is $\log{(0.99\dots)}$ negative or it is $ 0$?

Question

It is known that $0.99\dots =1$, but I'm afraid to say by substitution if it is allowed that $\log{(0.99\dots)}=0$ , then is it negative or equal $0$ ?

Answer 1

$0.9999...$ is exactly one. Is not an approximation or some other construct; the symbol $0.999...$ and the symbol $1$ are the exact same object, and you can write one or the other in any context

Answer 2

Another approach:

$S_n:= 9(1/10)+$

$9(1/10)^2 +9(1/10)^3...+9(1/10)^n.$

$\lim_{n \rightarrow \infty}S_n =1$,

the limit of the infinite geometric series with $r=(1/10)$.

Since $\log$ is a continuous function :

$\lim_{n \rightarrow \infty} \log(S_n) =$

$\log(\lim_{n \rightarrow \infty}S_n)=0.$