Is there a name for a function which is like $f(x)=-\log(-\log(x))$ where $0<x<1$?
Or, is there any name for this function $g(x)=x+\log(\frac{1}{x})$ where $0<x$?
Exchanging $x=-\log k$ in $g(x)$ gives $f(k)$ and I would like to know about those functions any deeper, but I am having trouble searching about them. If there are any specific name or related function that I can search for, I will be very glad to know.
On
this function completely depends on the base which we are taking for -log(x) because log is defined for positive values $$ -\log(x) > 0 $$ $$ \log(x) < 0 $$ as x lies between 0 to 1
then base of $\log(x)$ should lie between 1 to infinity(1 not included) this is true only when it is not mentioned that base is 'e'
Your first function corresponds to the inverse of the standard cumulative Gumbel distribution and occurs in extreme value statistics. In statistics, the inverse of a cumulative distribution is also called a quantile function. So, that makes it the standard Gumbel quantile function.
Other than that, I cannot see a use for devoting special attention to this function, in any case not at the analysis level, as it is just a composite function and the interesting function from that point of view is just the logarithm.