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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.

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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.

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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'

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    $\begingroup$ When it's not specified $\log = \ln$ $\endgroup$ Jul 2, 2023 at 15:40
  • $\begingroup$ @julio_es_sui_glace In my half-century of reading and working in various flavors of science and engineering, log was assumed to be base 10 and ln was natural log. Of course this is Math SE and that makes a difference - if no base is given, it is assumed to be $e$ no matter how log is spelled. I added a note at the top of the answer to try to address this. There may be something here of value to future readers. $\endgroup$
    – uhoh
    Jul 3, 2023 at 4:04
  • $\begingroup$ I agree with you, but I think it is quite common for the Anglo-Saxon math community tu use $\log$ instead of $\ln$. For me it was also the base $10$ log but I got used to it… Informaticians tend to use $\log$ as well since they only need it up to proportionality and for basis decomposition. $\endgroup$ Jul 3, 2023 at 8:27
  • $\begingroup$ ok i understood your point , i am learning how to use math SE so i dont know what exactly to do $\endgroup$ Jul 3, 2023 at 13:46

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